@article{linder2022deciphering,
  title={Deciphering the impact of genetic variation on human polyadenylation using APARENT2},
  author={Linder, Johannes and Koplik, Samantha E and Kundaje, Anshul and Seelig, Georg},
  journal={Genome biology},
  volume={23},
  number={1},
  pages={232},
  year={2022},
  publisher={Springer}
}

@article{ambig_manip_spirtes_2004,
    author = {Spirtes, Peter and Scheines, Richard},
    title = {Causal inference of ambiguous manipulations},
    journal = {Philosophy of Science},
    year = {2004}
}

@article{causal_ground_faller_2024,
    author = {Faller, Philipp and Vankadara, Leena and Mastakouri, Atalanti and Locatello, Francesco and Janzing, Dominik},
    title = {Self-Compatibility: Evaluating Causal Discovery without Ground Truth},
    journal = {International Conference on Artifical Intellgience and Statistics },
    year = {2024}
}

@article{prosemary_learning_structure,
  title={Learning to induce causal structure},
  author={Ke, Nan and Chiappa, Silvia and Wang, Jane and Bornschein, Jorg and Goyal, Anirudh and Rey, Melanie and Weber, Theophane and Botvinick, Matthew and Mozer, Michael and Rezende, Danilo},
  journal={International Conference on Learning Representations},
  year={2023}
}

@article{phillip_neural_cyclic,
  title={Efficient Neural Causal Discovery without Acyclicity Constraints},
  author={Lippe, Phillip and Cohen, Taco and Gavves, Efstratios},
  journal={International Conference on Learning Representations},
  year={2022}
}

@article{proxy_pearl_2014,
  title={Measurement bias and effect
restoration in causal inference},
 author={Kuroki, Manabu and Pearl, Judea},
  year={2014},
url = {https://academic.oup.com/biomet/article-abstract/101/2/423/194920}
}

@article{proxy_miao_2018,
  title={Identifying causal effects with proxy variables of an unmeasured confounder},
 author={Miao, Wang and Geng, Zhi and Tchetgen Tchetgen, Eric},
  year={2018},
url = {https://academic.oup.com/biomet/article-abstract/105/4/987/5073056?redirectedFrom=fulltext}
}


@article{raskutti_2013,
  title={Learning directed acyclic graphs based on sparsest permutations},
  author={Raskutti, Garvesh and Uhler, Caroline},
  year={2013},
journal = {STAT}
}

@article{suj_2024,
  title={Hybrid Top-Down Global Causal Discovery with Local Search for Linear and Nonlinear Additive Noise Models},
  author={Hiremath, Sujai and Maasch, Jacqueline and Gao, Mengxiao and Ghosal, Promit and Gan, Kyra},
note = {https://arxiv.org/abs/2405.14496},
journal = {Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS)},
year = {2024}
}


@inproceedings{caps_xu2024ordering,
  title={Ordering-Based Causal Discovery for Linear and Nonlinear Relations},
  author={Xu, Zhuopeng and Li, Yujie and Liu, Cheng and Gui, Ning},
  booktitle={Proceedings of the 37th Conference on Neural Information Processing Systems (NeurIPS)},
  year={2024}
}


@article{proxy_discovery_2024,
  title={Causal Discovery via Conditional Independence Testing with Proxy Variables},
  author={Liu, Mingzhou and Sun, Xinwei and Qiao, Yu and Wang, Yizhou},
  year={2024},
url = {https://arxiv.org/pdf/2305.05281}
}




@article{wang2022single,
  title={Single-cell multiome of the human retina and deep learning nominate causal variants in complex eye diseases},
  author={Wang, Sean K and Nair, Surag and Li, Rui and Kraft, Katerina and Pampari, Anusri and Patel, Aman and Kang, Joyce B and Luong, Christy and Kundaje, Anshul and Chang, Howard Y},
  journal={Cell genomics},
  volume={2},
  number={8},
  year={2022},
  publisher={Elsevier}
}

@article{wong2021decoding,
  title={Decoding disease: from genomes to networks to phenotypes},
  author={Wong, Aaron K and Sealfon, Rachel SG and Theesfeld, Chandra L and Troyanskaya, Olga G},
  journal={Nature Reviews Genetics},
  volume={22},
  number={12},
  pages={774--790},
  year={2021},
  publisher={Nature Publishing Group UK London}
}

@article{gan2021greedy,
  title={Greedy Approximation Algorithms for Active Sequential Hypothesis Testing},
  author={Gan, Kyra and Jia, Su and Li, Andrew},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={5012--5024},
  year={2021}
}

@article{radovic_minimum_2017,
	title = {Minimum redundancy maximum relevance feature selection approach for temporal gene expression data},
	volume = {18},
	issn = {1471-2105},
	url = {https://bmcbioinformatics.biomedcentral.com/articles/10.1186/s12859-016-1423-9},
	doi = {10.1186/s12859-016-1423-9},
	abstract = {Background: Feature selection, aiming to identify a subset of features among a possibly large set of features that are relevant for predicting a response, is an important preprocessing step in machine learning. In gene expression studies this is not a trivial task for several reasons, including potential temporal character of data. However, most feature selection approaches developed for microarray data cannot handle multivariate temporal data without previous data flattening, which results in loss of temporal information. We propose a temporal minimum redundancy - maximum relevance (TMRMR) feature selection approach, which is able to handle multivariate temporal data without previous data flattening. In the proposed approach we compute relevance of a gene by averaging F-statistic values calculated across individual time steps, and we compute redundancy between genes by using a dynamical time warping approach.
Results: The proposed method is evaluated on three temporal gene expression datasets from human viral challenge studies. Obtained results show that the proposed method outperforms alternatives widely used in gene expression studies. In particular, the proposed method achieved improvement in accuracy in 34 out of 54 experiments, while the other methods outperformed it in no more than 4 experiments.
Conclusion: We developed a filter-based feature selection method for temporal gene expression data based on maximum relevance and minimum redundancy criteria. The proposed method incorporates temporal information by combining relevance, which is calculated as an average F-statistic value across different time steps, with redundancy, which is calculated by employing dynamical time warping approach. As evident in our experiments, incorporating the temporal information into the feature selection process leads to selection of more discriminative features.},
	language = {en},
	number = {1},
	urldate = {2022-04-27},
	journal = {BMC Bioinformatics},
	author = {Radovic, Milos and Ghalwash, Mohamed and Filipovic, Nenad and Obradovic, Zoran},
	month = dec,
	year = {2017},
	pages = {9},
	file = {Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:/Users/jacquelinemaasch/Zotero/storage/Q64EPPYM/Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:application/pdf;Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:/Users/jacquelinemaasch/Zotero/storage/8CMR4VNZ/Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:application/pdf;Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:/Users/jacquelinemaasch/Zotero/storage/WYDFGCDR/Radovic et al. - 2017 - Minimum redundancy maximum relevance feature selec.pdf:application/pdf},
}

@book{hernan_causal_2020,
	title = {Causal {Inference}: {What} {If}},
	language = {en},
	author = {Hernán, Miguel A and Robins, James M},
	year = {2020},
	file = {Hernán and Robins - Causal Inference What If.pdf:/Users/jacquelinemaasch/Zotero/storage/WICT9UI8/Hernán and Robins - Causal Inference What If.pdf:application/pdf},
}

@article{kaddour_causal_2022,
	title = {Causal {Machine} {Learning}: {A} {Survey} and {Open} {Problems}},
	abstract = {Causal Machine Learning (CausalML) is an umbrella term for machine learning methods that formalize the data-generation process as a structural causal model (SCM). This perspective enables us to reason about the eﬀects of changes to this process (interventions) and what would have happened in hindsight (counterfactuals). We categorize work in CausalML into ﬁve groups according to the problems they address: (1) causal supervised learning, (2) causal generative modeling, (3) causal explanations, (4) causal fairness, and (5) causal reinforcement learning. We systematically compare the methods in each category and point out open problems. Further, we review data-modality-speciﬁc applications in computer vision, natural language processing, and graph representation learning. Finally, we provide an overview of causal benchmarks and a critical discussion of the state of this nascent ﬁeld, including recommendations for future work.},
	language = {en},
	author = {Kaddour, Jean and Lynch, Aengus and Liu, Qi and Kusner, Matt J and Silva, Ricardo},
	year = {2022},
	pages = {191},
	file = {Kaddour et al. - Causal Machine Learning A Survey and Open Problem.pdf:/Users/jacquelinemaasch/Zotero/storage/F8BZ98BU/Kaddour et al. - Causal Machine Learning A Survey and Open Problem.pdf:application/pdf},
}

@article{wang_blessings_2019,
	title = {The {Blessings} of {Multiple} {Causes}},
	volume = {114},
	issn = {0162-1459, 1537-274X},
	url = {https://www.tandfonline.com/doi/full/10.1080/01621459.2019.1686987},
	doi = {10.1080/01621459.2019.1686987},
	abstract = {Causal inference from observational data often assumes “ignorability,” that all confounders are observed. This assumption is standard yet untestable. However, many scientiﬁc studies involve multiple causes, different variables whose effects are simultaneously of interest. We propose the deconfounder, an algorithm that combines unsupervised machine learning and predictive model checking to perform causal inference in multiple-cause settings. The deconfounder infers a latent variable as a substitute for unobserved confounders and then uses that substitute to perform causal inference. We develop theory for the deconfounder, and show that it requires weaker assumptions than classical causal inference. We analyze its performance in three types of studies: semi-simulated data around smoking and lung cancer, semi-simulated data around genome-wide association studies, and a real dataset about actors and movie revenue. The deconfounder provides a checkable approach to estimating closer-to-truth causal effects.},
	language = {en},
	number = {528},
	urldate = {2022-08-29},
	journal = {Journal of the American Statistical Association},
	author = {Wang, Yixin and Blei, David M.},
	month = oct,
	year = {2019},
	pages = {1574--1596},
	file = {Wang and Blei - 2019 - The Blessings of Multiple Causes.pdf:/Users/jacquelinemaasch/Zotero/storage/Q2VXQYSX/Wang and Blei - 2019 - The Blessings of Multiple Causes.pdf:application/pdf},
}

@article{prosperi_causal_2020,
	title = {Causal inference and counterfactual prediction in machine learning for actionable healthcare},
	volume = {2},
	issn = {2522-5839},
	url = {http://www.nature.com/articles/s42256-020-0197-y},
	doi = {10.1038/s42256-020-0197-y},
	language = {en},
	number = {7},
	urldate = {2022-09-01},
	journal = {Nature Machine Intelligence},
	author = {Prosperi, Mattia and Guo, Yi and Sperrin, Matt and Koopman, James S. and Min, Jae S. and He, Xing and Rich, Shannan and Wang, Mo and Buchan, Iain E. and Bian, Jiang},
	month = jul,
	year = {2020},
	pages = {369--375},
	file = {Prosperi et al. - 2020 - Causal inference and counterfactual prediction in .pdf:/Users/jacquelinemaasch/Zotero/storage/JY634RPY/Prosperi et al. - 2020 - Causal inference and counterfactual prediction in .pdf:application/pdf},
}

@article{shen_challenges_2020,
	title = {Challenges and {Opportunities} with {Causal} {Discovery} {Algorithms}: {Application} to {Alzheimer}’s {Pathophysiology}},
	volume = {10},
	number = {2975},
	journal = {Scientific Reports},
	author = {Shen, Xinpeng and Ma, Sisi and Vemuri, Prashanthi and Simon, Gyorgy and {The Alzheimer’s Disease neuroimaging initiative}},
	year = {2020},
	file = {Shen - Challenges and Opportunities with Causal Discovery.pdf:/Users/jacquelinemaasch/Zotero/storage/8P4LQ98R/Shen - Challenges and Opportunities with Causal Discovery.pdf:application/pdf},
}

@misc{lippe_efficient_2022,
	title = {Efficient {Neural} {Causal} {Discovery} without {Acyclicity} {Constraints}},
	url = {http://arxiv.org/abs/2107.10483},
	abstract = {Learning the structure of a causal graphical model using both observational and interventional data is a fundamental problem in many scientiﬁc ﬁelds. A promising direction is continuous optimization for score-based methods, which, however, require constrained optimization to enforce acyclicity or lack convergence guarantees. In this paper, we present ENCO, an efﬁcient structure learning method for directed, acyclic causal graphs leveraging observational and interventional data. ENCO formulates the graph search as an optimization of independent edge likelihoods, with the edge orientation being modeled as a separate parameter. Consequently, we provide for ENCO convergence guarantees when interventions on all variables are available, without having to constrain the score function with respect to acyclicity. In experiments, we show that ENCO can efﬁciently recover graphs with hundreds of nodes, an order of magnitude larger than what was previously possible, while handling deterministic variables and discovering latent confounders.},
	language = {en},
	urldate = {2022-09-13},
	publisher = {arXiv},
	author = {Lippe, Phillip and Cohen, Taco and Gavves, Efstratios},
	month = feb,
	year = {2022},
	note = {arXiv:2107.10483 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Lippe et al. - 2022 - Efficient Neural Causal Discovery without Acyclici.pdf:/Users/jacquelinemaasch/Zotero/storage/UAJXXZRL/Lippe et al. - 2022 - Efficient Neural Causal Discovery without Acyclici.pdf:application/pdf},
}

@misc{annadani_variational_2021,
	title = {Variational {Causal} {Networks}: {Approximate} {Bayesian} {Inference} over {Causal} {Structures}},
	shorttitle = {Variational {Causal} {Networks}},
	url = {http://arxiv.org/abs/2106.07635},
	abstract = {Learning the causal structure that underlies data is a crucial step towards robust real-world decision making. The majority of existing work in causal inference focuses on determining a single directed acyclic graph (DAG) or a Markov equivalence class thereof. However, a crucial aspect to acting intelligently upon the knowledge about causal structure which has been inferred from ﬁnite data demands reasoning about its uncertainty. For instance, planning interventions to ﬁnd out more about the causal mechanisms that govern our data requires quantifying epistemic uncertainty over DAGs. While Bayesian causal inference allows to do so, the posterior over DAGs becomes intractable even for a small number of variables. Aiming to overcome this issue, we propose a form of variational inference over the graphs of Structural Causal Models (SCMs). To this end, we introduce a parametric variational family modelled by an autoregressive distribution over the space of discrete DAGs. Its number of parameters does not grow exponentially with the number of variables and can be tractably learned by maximising an Evidence Lower Bound (ELBO). In our experiments, we demonstrate that the proposed variational posterior is able to provide a good approximation of the true posterior.},
	language = {en},
	urldate = {2022-09-13},
	publisher = {arXiv},
	author = {Annadani, Yashas and Rothfuss, Jonas and Lacoste, Alexandre and Scherrer, Nino and Goyal, Anirudh and Bengio, Yoshua and Bauer, Stefan},
	month = jun,
	year = {2021},
	note = {arXiv:2106.07635 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Annadani et al. - 2021 - Variational Causal Networks Approximate Bayesian .pdf:/Users/jacquelinemaasch/Zotero/storage/HNDZHRRV/Annadani et al. - 2021 - Variational Causal Networks Approximate Bayesian .pdf:application/pdf},
}

@inproceedings{cundy_bcd_2021,
	title = {{BCD} {Nets}: {Scalable} {Variational} {Approaches} for {Bayesian} {Causal} {Discovery}},
	abstract = {A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identiﬁable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizing a linear-Gaussian SEM. Developing a full Bayesian posterior over DAGs is challenging due to the the discrete and combinatorial nature of graphs. We analyse key design choices for scalable VI over DAGs, such as 1) the parametrization of DAGs via an expressive variational family, 2) a continuous relaxation that enables low-variance stochastic optimization, and 3) suitable priors over the latent variables. We provide a series of experiments on real and synthetic data showing that BCD Nets outperform maximum-likelihood methods on standard causal discovery metrics such as structural Hamming distance in low data regimes.},
	language = {en},
	booktitle = {35th {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Cundy, Chris and Grover, Aditya and Ermon, Stefano},
	year = {2021},
	file = {Cundy et al. - BCD Nets Scalable Variational Approaches for Baye.pdf:/Users/jacquelinemaasch/Zotero/storage/H72SW2PM/Cundy et al. - BCD Nets Scalable Variational Approaches for Baye.pdf:application/pdf},
}

@inproceedings{lorch_dibs_2021,
	title = {{DiBS}: {Differentiable} {Bayesian} {Structure} {Learning}},
	language = {en},
	booktitle = {35th {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Lorch, Lars and Schölkopf, Bernhard and Rothfuss, Jonas and Krause, Andreas},
	year = {2021},
	file = {Lorch et al. - DiBS Differentiable Bayesian Structure Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/DEDVQSAH/Lorch et al. - DiBS Differentiable Bayesian Structure Learning.pdf:application/pdf},
}

@inproceedings{sinha_using_2021,
	title = {Using a {General} {Prior} {Knowledge} {Graph} to {Improve} {Data}-{Driven} {Causal} {Network} {Learning}},
	url = {https://proceedings.aaai-make.info/AAAI-MAKE-PROCEEDINGS-2021/paper31.pdf},
	abstract = {We describe a method “Kg2Causal” for using a large-scale, general-purpose biomedical knowledge graph as a prior for data-driven causal network structure learning. Given a set of observed nodes in a dataset, and some relationship edges between the nodes derived from a knowledge graph, Kg2Causal uses the knowledge graph-derived edges to guide the data-driven inference of a causal Bayesian network. We tested Kg2Causal on several real-world biological datasets with known ground-truth networks and demonstrate improvement in network learning accuracy, relative to a baseline of an uninformative network structure prior. We also demonstrate the application of our method if data are collected under different experimental conditions including interventions on the observed variables.},
	language = {en},
	booktitle = {Proceedings of the {AAAI} 2021 {Spring} {Symposium} on {Combining} {Machine} {Learning} and {Knowledge} {Engineering}},
	author = {Sinha, Meghamala and Ramsey, Stephen A},
	year = {2021},
	file = {Sinha and Ramsey - Using a General Prior Knowledge Graph to Improve D.pdf:/Users/jacquelinemaasch/Zotero/storage/F9WP9IJ7/Sinha and Ramsey - Using a General Prior Knowledge Graph to Improve D.pdf:application/pdf},
}

@misc{halpern_causes_2005,
	title = {Causes and {Explanations}: {A} {Structural}-{Model} {Approach}, {Part} {I}: {Causes}},
	shorttitle = {Causes and {Explanations}},
	url = {http://arxiv.org/abs/cs/0011012},
	abstract = {We propose a new deﬁnition of actual causes, using structural equations to model counterfactuals. We show that the deﬁnition yields a plausible and elegant account of causation that handles well examples which have caused problems for other deﬁnitions and resolves major difﬁculties in the traditional account.},
	language = {en},
	urldate = {2022-09-13},
	publisher = {arXiv},
	author = {Halpern, Joseph Y. and Pearl, Judea},
	month = nov,
	year = {2005},
	note = {arXiv:cs/0011012},
	keywords = {Computer Science - Artificial Intelligence, I.2.4},
	file = {Halpern and Pearl - 2005 - Causes and Explanations A Structural-Model Approa.pdf:/Users/jacquelinemaasch/Zotero/storage/9N6RVDCA/Halpern and Pearl - 2005 - Causes and Explanations A Structural-Model Approa.pdf:application/pdf},
}

@article{zheng_dags_nodate,
	title = {{DAGs} with {NO} {TEARS}: {Continuous} {Optimization} for {Structure} {Learning}},
	abstract = {Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: we formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efﬁciently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree.},
	language = {en},
	author = {Zheng, Xun and Aragam, Bryon and Ravikumar, Pradeep K and Xing, Eric P},
	pages = {12},
	file = {Zheng et al. - DAGs with NO TEARS Continuous Optimization for St.pdf:/Users/jacquelinemaasch/Zotero/storage/A26LQCLH/Zheng et al. - DAGs with NO TEARS Continuous Optimization for St.pdf:application/pdf},
}

@inproceedings{hasan_kcrl_2022,
	title = {{KCRL}: {A} {Prior} {Knowledge} {Based} {Causal} {Discovery} {Framework} with {Reinforcement} {Learning}},
	abstract = {Causal discovery is an important problem in many sciences that enables us to estimate causal relationships from observational data. Particularly, in the healthcare domain, it can guide practitioners in making informed clinical decisions. Several causal discovery approaches have been developed over the last few decades. The success of these approaches mostly rely on a large number of data samples. In practice, however, an infinite amount of data is never available. Fortunately, often we have some prior knowledge available from the problem domain. Particularly, in healthcare settings, we often have some prior knowledge such as expert opinions, prior RCTs, literature evidence, and systematic reviews about the clinical problem. This prior information can be utilized in a systematic way to address the data scarcity problem. However, most of the existing causal discovery approaches lack a systematic way to incorporate prior knowledge during the search process. Recent advances in reinforcement learning techniques can be explored to use prior knowledge as constraints by penalizing the agent for their violations. Therefore, in this work, we propose a framework KCRL 1 that utilizes the existing knowledge as a constraint to penalize the search process during causal discovery. This utilization of existing information during causal discovery reduces the graph search space and enables a faster convergence to the optimal causal mechanism. We evaluated our framework on benchmark synthetic and real datasets as well as on a real-life healthcare application. We also compared its performance with several baseline causal discovery methods. The experimental findings show that penalizing the search process for constraint violation yields better performance compared to existing approaches that do not utilize prior knowledge.},
	language = {en},
	booktitle = {Proceedings of {Machine} {Learning} {Research}},
	author = {Hasan, Uzma and Gani, Osman},
	year = {2022},
	file = {Hasan and Gani - 2022 - KCRL A Prior Knowledge Based Causal Discovery Fra.pdf:/Users/jacquelinemaasch/Zotero/storage/ZFK55SMX/Hasan and Gani - 2022 - KCRL A Prior Knowledge Based Causal Discovery Fra.pdf:application/pdf},
}

@article{spirtes_algorithm_1991,
	title = {An {Algorithm} for {Fast} {Recovery} of {Sparse} {Causal} {Graphs}},
	volume = {9},
	issn = {0894-4393, 1552-8286},
	url = {http://journals.sagepub.com/doi/10.1177/089443939100900106},
	doi = {10.1177/089443939100900106},
	abstract = {Previous asymptotically correct algorithms for recovering causal structure from sample probabilities have been limited even in sparse causal graphs to a few variables. We describe an asymptotically correct algorithm whose complexity for fixed graph connectivity increases polynomially in the number of vertices, and may in practice recover sparse graphs with several hundred variables. From sample data with n = 20,000, an implementation of the algorithm on a DECStation 3100 recovers the edges in a linear version of the ALARM network with 37 vertices and 46 edges. Fewer than 8\% of the undirected edges are incorrectly identified in the output. Without prior ordering information, the program also determines the direction of edges for the ALARM graph with an error rate of 14\%. Processing time is less than 10 seconds. Keywords DAGS, Causal Modelling.},
	language = {en},
	number = {1},
	urldate = {2022-09-15},
	journal = {Social Science Computer Review},
	author = {Spirtes, Peter and Glymour, Clark},
	month = apr,
	year = {1991},
	pages = {62--72},
	file = {Spirtes and Glymour - 1991 - An Algorithm for Fast Recovery of Sparse Causal Gr.pdf:/Users/jacquelinemaasch/Zotero/storage/LGC9SWX3/Spirtes and Glymour - 1991 - An Algorithm for Fast Recovery of Sparse Causal Gr.pdf:application/pdf},
}

@inproceedings{viinikka_towards_2020,
	title = {Towards {Scalable} {Bayesian} {Learning} of {Causal} {DAGs}},
	abstract = {We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efﬁcient approximate sampling from the graph posterior, provided that each node is assigned a small number K of candidate parents. We present algorithmic techniques to signiﬁcantly reduce the space and time requirements, which make the use of substantially larger values of K feasible. Furthermore, we investigate the problem of selecting the candidate parents per node so as to maximize the covered posterior mass. Finally, we combine our sampling method with a novel Bayesian approach for estimating causal effects in linear Gaussian DAG models. Numerical experiments demonstrate the performance of our methods in detecting ancestor–descendant relations, and in causal effect estimation our Bayesian method is shown to outperform previous approaches.},
	booktitle = {34th {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Viinikka, Jussi and Hyttinen, Antti},
	year = {2020},
	file = {Viinikka and Hyttinen - 2020 - Towards Scalable Bayesian Learning of Causal DAGs.pdf:/Users/jacquelinemaasch/Zotero/storage/UJSSJDGB/Viinikka and Hyttinen - 2020 - Towards Scalable Bayesian Learning of Causal DAGs.pdf:application/pdf},
}

@misc{dorn_doubly-validdoubly-sharp_2022,
	title = {Doubly-{Valid}/{Doubly}-{Sharp} {Sensitivity} {Analysis} for {Causal} {Inference} with {Unmeasured} {Confounding}},
	url = {http://arxiv.org/abs/2112.11449},
	abstract = {We consider the problem of constructing bounds on the average treatment eﬀect (ATE) when unmeasured confounders exist but have bounded inﬂuence. Speciﬁcally, we assume that omitted confounders could not change the odds of treatment for any unit by more than a ﬁxed factor. We derive the sharp partial identiﬁcation bounds implied by this assumption by leveraging distributionally robust optimization, and we propose estimators of these bounds with several novel robustness properties. The ﬁrst is double sharpness: our estimators consistently estimate the sharp ATE bounds when one of two nuisance parameters is misspeciﬁed and achieve semiparametric eﬃciency when all nuisance parameters are suitably consistent. The second is double validity: even when most nuisance parameters are misspeciﬁed, our estimators still provide valid but possibly conservative bounds for the ATE and our Wald conﬁdence intervals remain valid even when our estimators are not asymptotically normal. As a result, our estimators provide a highly credible method for sensitivity analysis of causal inferences.},
	language = {en},
	urldate = {2022-10-07},
	publisher = {arXiv},
	author = {Dorn, Jacob and Guo, Kevin and Kallus, Nathan},
	month = jul,
	year = {2022},
	note = {arXiv:2112.11449 [cs, econ, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology, Economics - Econometrics, Mathematics - Optimization and Control},
	file = {Dorn et al. - 2022 - Doubly-ValidDoubly-Sharp Sensitivity Analysis for.pdf:/Users/jacquelinemaasch/Zotero/storage/7B8Z32BG/Dorn et al. - 2022 - Doubly-ValidDoubly-Sharp Sensitivity Analysis for.pdf:application/pdf},
}
@article{addo2021exploring,
  title={Exploring nonlinearity on the CO2 emissions, economic production and energy use nexus: a causal discovery approach},
  author={Addo, Peter Martey and Manibialoa, Christelle and McIsaac, Florent},
  journal={Energy Reports},
  volume={7},
  pages={6196--6204},
  year={2021},
  publisher={Elsevier}
}
@article{manicka2023nonlinearity,
  author={Manicka, Santosh and Johnson, Karl and Levin, Michael and Karlebach, Guy},
  title={The nonlinearity of regulation in biological networks},
  journal={npj Systems Biology and Applications},
  volume={9},
  number={10},
  year={2023},
  month={April},
  doi={10.1038/s41540-023-00273-w}
}


@article{kontio2020estimating,
  author={Kontio, Juho A. J. and Rinta-aho, Marko J. and Sillanpää, Mikko J.},
  title={Estimating Linear and Nonlinear Gene Coexpression Networks by Semiparametric Neighborhood Selection},
  journal={Genetics},
  volume={215},
  number={3},
  pages={597--607},
  year={2020},
  month={July},
  doi={10.1534/genetics.120.303186}
}



@article{lee_causal_2022,
	title = {Causal determinants of postoperative length of stay in cardiac surgery using causal graphical learning},
	issn = {00225223},
	doi = {10.1016/j.jtcvs.2022.08.012},
	abstract = {Objective: We aimed to learn the causal determinants of postoperative length of stay in cardiac surgery patients undergoing isolated coronary artery bypass grafting or aortic valve replacement surgery.
Methods: For patients undergoing isolated coronary artery bypass grafting or isolated aortic valve replacement surgeries between 2011 and 2016, we used causal graphical modeling on electronic health record data. The Fast Causal Inference (FCI) algorithm from the Tetrad software was used on data to estimate a Partial Ancestral Graph (PAG) depicting direct and indirect causes of postoperative length of stay, given background clinical knowledge. Then, we used the latent variable intervention-calculus when the directed acyclic graph is absent (LV-IDA) algorithm to estimate strengths of causal effects of interest. Finally, we ran a linear regression for postoperative length of stay to contrast statistical associations with what was learned by our causal analysis.
Results: In our cohort of 2610 patients, the mean postoperative length of stay was 219 hours compared with the Society of Thoracic Surgeons 2016 national mean postoperative length of stay of approximately 168 hours. Most variables that clinicians believe to be predictors of postoperative length of stay were found to be causes, but some were direct (eg, age, diabetes, hematocrit, total operating time, and postoperative complications), and others were indirect (including gender, race, and operating surgeon). The strongest average causal effects on postoperative length of stay were exhibited by preoperative dialysis (209 hours); neuro-, pulmonary-, and infection-related postoperative complications (315 hours, 89 hours, and 131 hours, respectively); reintubation (61 hours); extubation in operating room (–47 hours); and total operating room duration (48 hours). Linear regression coefﬁcients diverged from causal effects in magnitude (eg, dialysis) and direction (eg, crossclamp time). pLOS Causal graphical learning uncovers insights that can reduce pLOS. CENTRAL MESSAGE Causal graphical modeling aims to discern indirect and direct causes of pLOS, not simply associations. This helps generate data-driven hypotheses for clinical protocol design. PERSPECTIVE Minimizing pLOS is fundamental for efﬁcient, high-quality care. Prior analyses (eg, ML/statistical regression) have not always distinguished associations from causal drivers of pLOS, leading to misdirected attempts to improve it. Using causal graphical modeling, we identify causal relationships underlying pLOS to target improvements in patient management. See Commentary on page XXX.
Conclusions: By using retrospective electronic health record data and background clinical knowledge, causal graphical modeling retrieved direct and indirect causes of postoperative length of stay and their relative strengths. These insights will be useful in designing clinical protocols and targeting improvements in patient management. (J Thorac Cardiovasc Surg 2022;-:1-10)},
	language = {en},
	urldate = {2022-10-07},
	journal = {The Journal of Thoracic and Cardiovascular Surgery},
	author = {Lee, Jaron J.R. and Srinivasan, Ranjani and Ong, Chin Siang and Alejo, Diane and Schena, Stefano and Shpitser, Ilya and Sussman, Marc and Whitman, Glenn J.R. and Malinsky, Daniel},
	month = aug,
	year = {2022},
	pages = {S002252232200900X},
	file = {Lee et al. - 2022 - Causal determinants of postoperative length of sta.pdf:/Users/jacquelinemaasch/Zotero/storage/2448FEKL/Lee et al. - 2022 - Causal determinants of postoperative length of sta.pdf:application/pdf},
}

@inproceedings{malinsky2016estimating,
  title={Estimating causal effects with ancestral graph Markov models},
  author={Malinsky, Daniel and Spirtes, Peter},
  booktitle={Proceedings of the Conference on Probabilistic Graphical Models (PGM)},
  pages={319--330},
  year={2016},
  organization={PMLR}
}


@article{tamer_partial_2010,
	title = {Partial {Identification} in {Econometrics}},
	volume = {2},
	issn = {1941-1383, 1941-1391},
	url = {https://www.annualreviews.org/doi/10.1146/annurev.economics.050708.143401},
	doi = {10.1146/annurev.economics.050708.143401},
	abstract = {Identification in econometric models maps prior assumptions and the data to information about a parameter of interest. The partial identification approach to inference recognizes that this process should not result in a binary answer that consists of whether the parameter is point identified. Rather, given the data, the partial identification approach characterizes the informational content of various assumptions by providing a menu of estimates, each based on different sets of assumptions, some of which are plausible and some of which are not. Of course, more assumptions beget more information, so stronger conclusions can be made at the expense of more assumptions. The partial identification approach advocates a more fluid view of identification and hence provides the empirical researcher with methods to help study the spectrum of information that we can harness about a parameter of interest using a menu of assumptions. This approach links conclusions drawn from various empirical models to sets of assumptions made in a transparent way. It allows researchers to examine the informational content of their assumptions and their impacts on the inferences made. Naturally, with finite sample sizes, this approach leads to statistical complications, as one needs to deal with characterizing sampling uncertainty in models that do not point identify a parameter. Therefore, new methods for inference are developed. These methods construct confidence sets for partially identified parameters, and confidence regions for sets of parameters, or identifiable sets.},
	language = {en},
	number = {1},
	urldate = {2022-10-07},
	journal = {Annual Review of Economics},
	author = {Tamer, Elie},
	month = sep,
	year = {2010},
	pages = {167--195},
	file = {Tamer - 2010 - Partial Identification in Econometrics.pdf:/Users/jacquelinemaasch/Zotero/storage/MNWYEV4G/Tamer - 2010 - Partial Identification in Econometrics.pdf:application/pdf},
}

@book{chernozhukov_applied_2022,
	title = {Applied {Causal} {Inference} {Powered} by {ML} and {AI}},
	url = {https://github.com/fmarotta/kaobook},
	publisher = {Authors for Online Distribution},
	author = {Chernozhukov, Victor and Hansen, Christian and Kallus, Nathan and Spindler, Martin and Syrgkanis, Vasilis},
	year = {2022},
	file = {Chernozhukov et al. - 2022 - Applied Causal Inference Powered by ML and AI.pdf:/Users/jacquelinemaasch/Zotero/storage/S65GI3UJ/Chernozhukov et al. - 2022 - Applied Causal Inference Powered by ML and AI.pdf:application/pdf},
}

@article{malinsky_causal_2018,
	title = {Causal discovery algorithms: {A} practical guide},
	volume = {13},
	issn = {17479991},
	shorttitle = {Causal discovery algorithms},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/phc3.12470},
	doi = {10.1111/phc3.12470},
	abstract = {Many investigations into the world, including philosophical ones, aim to discover causal knowledge, and many experimental methods have been developed to assist in causal discovery. More recently, algorithms have emerged that can also learn causal structure from purely or mostly observational data, as well as experimental data. These methods have started to be applied in various philosophical contexts, such as debates about our concepts of free will and determinism. This paper provides a “user's guide” to these methods, though not in the sense of specifying exact button presses in a software package. Instead, we explain the larger “pipeline” within which these methods are used and discuss key steps in moving from initial research idea to validated causal structure.},
	language = {en},
	number = {1},
	urldate = {2022-10-17},
	journal = {Philosophy Compass},
	author = {Malinsky, Daniel and Danks, David},
	month = jan,
	year = {2018},
	pages = {e12470},
	file = {Malinsky and Danks - 2018 - Causal discovery algorithms A practical guide.pdf:/Users/jacquelinemaasch/Zotero/storage/JBDQ9E29/Malinsky and Danks - 2018 - Causal discovery algorithms A practical guide.pdf:application/pdf},
}



@article{scheines_introduction_1997,
	title = {An introduction to causal inference},
	language = {en},
	author = {Scheines, Richard},
	year = {1997},
	pages = {16},
	file = {Scheines - 1997 - An introduction to causal inference.pdf:/Users/jacquelinemaasch/Zotero/storage/8GY2EA6Q/Scheines - 1997 - An introduction to causal inference.pdf:application/pdf},
}

@misc{scheines_d-separation_nodate,
	title = {D-{Separation}},
	url = {https://www.andrew.cmu.edu/user/scheines/tutor/d-sep.html},
	author = {Scheines, Richard},
	file = {Scheines - D-Separation.pdf:/Users/jacquelinemaasch/Zotero/storage/QL29AQHX/Scheines - D-Separation.pdf:application/pdf},
}

@article{teyssier2012ordering,
  title={Ordering-based search: A simple and effective algorithm for learning Bayesian networks},
  author={Teyssier, Marc and Koller, Daphne},
  journal={arXiv preprint arXiv:1207.1429},
  year={2012}
}



@article{glymour_review_2019,
	title = {Review of {Causal} {Discovery} {Methods} {Based} on {Graphical} {Models}},
	volume = {10},
	issn = {1664-8021},
	doi = {10.3389/fgene.2019.00524},
	abstract = {A fundamental task in various disciplines of science, including biology, is to ﬁnd underlying causal relations and make use of them. Causal relations can be seen if interventions are properly applied; however, in many cases they are difﬁcult or even impossible to conduct. It is then necessary to discover causal relations by analyzing statistical properties of purely observational data, which is known as causal discovery or causal structure search. This paper aims to give a introduction to and a brief review of the computational methods for causal discovery that were developed in the past three decades, including constraint-based and score-based methods and those based on functional causal models, supplemented by some illustrations and applications.},
	language = {en},
	urldate = {2022-11-01},
	journal = {Frontiers in Genetics},
	author = {Glymour, Clark and Zhang, Kun and Spirtes, Peter},
	month = jun,
	year = {2019},
	pages = {524},
	file = {Glymour et al. - 2019 - Review of Causal Discovery Methods Based on Graphi.pdf:/Users/jacquelinemaasch/Zotero/storage/CAK5QASA/Glymour et al. - 2019 - Review of Causal Discovery Methods Based on Graphi.pdf:application/pdf},
}

@article{eberhardt_introduction_2017,
	title = {Introduction to the foundations of causal discovery},
	volume = {3},
	issn = {2364-415X, 2364-4168},
	url = {http://link.springer.com/10.1007/s41060-016-0038-6},
	doi = {10.1007/s41060-016-0038-6},
	abstract = {This article presents an overview of several known approaches to causal discovery. It is organized by relating the different fundamental assumptions that the methods depend on. The goal is to indicate that for a large variety of different settings the assumptions necessary and sufﬁcient for causal discovery are now well understood.},
	language = {en},
	number = {2},
	urldate = {2022-11-01},
	journal = {International Journal of Data Science and Analytics},
	author = {Eberhardt, Frederick},
	month = mar,
	year = {2017},
	pages = {81--91},
	file = {Eberhardt - 2017 - Introduction to the foundations of causal discover.pdf:/Users/jacquelinemaasch/Zotero/storage/UBT6TM2A/Eberhardt - 2017 - Introduction to the foundations of causal discover.pdf:application/pdf},
}

@book{jonas_peters_elements_2017,
	address = {Cambridge, Massachusetts},
	title = {Elements of causal inference: foundations and learning algorithms},
	isbn = {978-0-262-03731-0},
	shorttitle = {Elements of causal inference},
	url = {https://mitpress.mit.edu/9780262037310/elements-of-causal-inference/},
	publisher = {The MIT Press},
	author = {{Jonas Peters} and {Dominik Janzing} and {Bernhard Scholkopf}},
	year = {2017},
	file = {Jonas Peters et al. - 2017 - Elements of causal inference foundations and lear.pdf:/Users/jacquelinemaasch/Zotero/storage/65LMEKZ2/Jonas Peters et al. - 2017 - Elements of causal inference foundations and lear.pdf:application/pdf},
}

@misc{deleu_bayesian_2022,
	title = {Bayesian {Structure} {Learning} with {Generative} {Flow} {Networks}},
	url = {http://arxiv.org/abs/2202.13903},
	abstract = {In Bayesian structure learning, we are interested in inferring a distribution over the directed acyclic graph (DAG) structure of Bayesian networks, from data. Deﬁning such a distribution is very challenging, due to the combinatorially large sample space, and approximations based on MCMC are often required. Recently, a novel class of probabilistic models, called Generative Flow Networks (GFlowNets), have been introduced as a general framework for generative modeling of discrete and composite objects, such as graphs. In this work, we propose to use a GFlowNet as an alternative to MCMC for approximating the posterior distribution over the structure of Bayesian networks, given a dataset of observations. Generating a sample DAG from this approximate distribution is viewed as a sequential decision problem, where the graph is constructed one edge at a time, based on learned transition probabilities. Through evaluation on both simulated and real data, we show that our approach, called DAG-GFlowNet, provides an accurate approximation of the posterior over DAGs, and it compares favorably against other methods based on MCMC or variational inference.},
	language = {en},
	urldate = {2022-11-07},
	publisher = {arXiv},
	author = {Deleu, Tristan and Góis, António and Emezue, Chris and Rankawat, Mansi and Lacoste-Julien, Simon and Bauer, Stefan and Bengio, Yoshua},
	month = jun,
	year = {2022},
	note = {arXiv:2202.13903 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Deleu et al. - 2022 - Bayesian Structure Learning with Generative Flow N.pdf:/Users/jacquelinemaasch/Zotero/storage/5DKIN3VJ/Deleu et al. - 2022 - Bayesian Structure Learning with Generative Flow N.pdf:application/pdf},
}

@misc{bengio_gflownet_2022,
	title = {{GFlowNet} {Foundations}},
	url = {http://arxiv.org/abs/2111.09266},
	abstract = {Generative Flow Networks (GFlowNets) have been introduced as a method to sample a diverse set of candidates in an active learning context, with a training objective that makes them approximately sample in proportion to a given reward function. In this paper, we show a number of additional theoretical properties of GFlowNets. They can be used to estimate joint probability distributions and the corresponding marginal distributions where some variables are unspeciﬁed and, of particular interest, can represent distributions over composite objects like sets and graphs. GFlowNets amortize the work typically done by computationally expensive MCMC methods in a single but trained generative pass. They could also be used to estimate partition functions and free energies, conditional probabilities of supersets (supergraphs) given a subset (subgraph), as well as marginal distributions over all supersets (supergraphs) of a given set (graph). We introduce variations enabling the estimation of entropy and mutual information, sampling from a Pareto frontier, connections to reward-maximizing policies, and extensions to stochastic environments, continuous actions and modular energy functions.},
	language = {en},
	urldate = {2022-11-07},
	publisher = {arXiv},
	author = {Bengio, Yoshua and Lahlou, Salem and Deleu, Tristan and Hu, Edward J. and Tiwari, Mo and Bengio, Emmanuel},
	month = aug,
	year = {2022},
	note = {arXiv:2111.09266 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Bengio et al. - 2022 - GFlowNet Foundations.pdf:/Users/jacquelinemaasch/Zotero/storage/2PLA8EB6/Bengio et al. - 2022 - GFlowNet Foundations.pdf:application/pdf},
}

@misc{bengio_flow_2021,
	title = {Flow {Network} based {Generative} {Models} for {Non}-{Iterative} {Diverse} {Candidate} {Generation}},
	url = {http://arxiv.org/abs/2106.04399},
	abstract = {This paper is about the problem of learning a stochastic policy for generating an object (like a molecular graph) from a sequence of actions, such that the probability of generating an object is proportional to a given positive reward for that object. Whereas standard return maximization tends to converge to a single return-maximizing sequence, there are cases where we would like to sample a diverse set of high-return solutions. These arise, for example, in black-box function optimization when few rounds are possible, each with large batches of queries, where the batches should be diverse, e.g., in the design of new molecules. One can also see this as a problem of approximately converting an energy function to a generative distribution. While MCMC methods can achieve that, they are expensive and generally only perform local exploration. Instead, training a generative policy amortizes the cost of search during training and yields to fast generation. Using insights from Temporal Difference learning, we propose GFlowNet, based on a view of the generative process as a ﬂow network, making it possible to handle the tricky case where different trajectories can yield the same ﬁnal state, e.g., there are many ways to sequentially add atoms to generate some molecular graph. We cast the set of trajectories as a ﬂow and convert the ﬂow consistency equations into a learning objective, akin to the casting of the Bellman equations into Temporal Difference methods. We prove that any global minimum of the proposed objectives yields a policy which samples from the desired distribution, and demonstrate the improved performance and diversity of GFlowNet on a simple domain where there are many modes to the reward function, and on a molecule synthesis task.},
	language = {en},
	urldate = {2022-11-07},
	publisher = {arXiv},
	author = {Bengio, Emmanuel and Jain, Moksh and Korablyov, Maksym and Precup, Doina and Bengio, Yoshua},
	month = nov,
	year = {2021},
	note = {arXiv:2106.04399 [cs]},
	keywords = {Computer Science - Machine Learning},
	file = {Bengio et al. - 2021 - Flow Network based Generative Models for Non-Itera.pdf:/Users/jacquelinemaasch/Zotero/storage/TY8UVWSL/Bengio et al. - 2021 - Flow Network based Generative Models for Non-Itera.pdf:application/pdf},
}

@misc{wang_causal_2021,
	title = {Causal {Discovery} with {Unobserved} {Confounding} and non-{Gaussian} {Data}},
	url = {http://arxiv.org/abs/2007.11131},
	abstract = {We consider recovering causal structure from multivariate observational data. We assume the data arise from a linear structural equation model (SEM) in which the idiosyncratic errors are allowed to be dependent in order to capture possible latent confounding. Each SEM can be represented by a graph where vertices represent observed variables, directed edges represent direct causal eﬀects, and bidirected edges represent dependence among error terms. Speciﬁcally, we assume that the true model corresponds to a bow-free acyclic path diagram; i.e., a graph that has at most one edge between any pair of nodes and is acyclic in the directed part. We show that when the errors are non-Gaussian, the exact causal structure encoded by such a graph, and not merely an equivalence class, can be recovered from observational data. The method we propose for this purpose uses estimates of suitable moments, but, in contrast to previous results, does not require specifying the number of latent variables a priori. We also characterize the output of our procedure when the assumptions are violated and the true graph is acyclic, but not bow-free. We illustrate the eﬀectiveness of our procedure in simulations and an application to an ecology data set.},
	language = {en},
	urldate = {2022-11-07},
	publisher = {arXiv},
	author = {Wang, Y. Samuel and Drton, Mathias},
	month = nov,
	year = {2021},
	note = {arXiv:2007.11131 [stat]},
	keywords = {Statistics - Methodology},
	file = {Wang and Drton - 2021 - Causal Discovery with Unobserved Confounding and n.pdf:/Users/jacquelinemaasch/Zotero/storage/RDF8KJ5J/Wang and Drton - 2021 - Causal Discovery with Unobserved Confounding and n.pdf:application/pdf},
}

@misc{vowels_ya_2021,
	title = {D'ya like {DAGs}? {A} {Survey} on {Structure} {Learning} and {Causal} {Discovery}},
	shorttitle = {D'ya like {DAGs}?},
	url = {http://arxiv.org/abs/2103.02582},
	abstract = {Causal reasoning is a crucial part of science and human intelligence. In order to discover causal relationships from data, we need structure discovery methods. We provide a review of background theory and a survey of methods for structure discovery. We primarily focus on modern, continuous optimization methods, and provide reference to further resources such as benchmark datasets and software packages. Finally, we discuss the assumptive leap required to take us from structure to causality.},
	language = {en},
	urldate = {2022-11-08},
	publisher = {arXiv},
	author = {Vowels, Matthew J. and Camgoz, Necati Cihan and Bowden, Richard},
	month = mar,
	year = {2021},
	note = {arXiv:2103.02582 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology},
	file = {Vowels et al. - 2021 - D'ya like DAGs A Survey on Structure Learning and.pdf:/Users/jacquelinemaasch/Zotero/storage/RQT6EMYA/Vowels et al. - 2021 - D'ya like DAGs A Survey on Structure Learning and.pdf:application/pdf},
}

@article{wang_tractable_nodate,
	title = {Tractable {Uncertainty} for {Structure} {Learning}},
	abstract = {Bayesian structure learning allows one to capture uncertainty over the causal directed acyclic graph (DAG) responsible for generating given data. In this work, we present Tractable Uncertainty for STructure learning (TRUST), a framework for approximate posterior inference that relies on probabilistic circuits as the representation of our posterior belief. In contrast to sample-based posterior approximations, our representation can capture a much richer space of DAGs, while also being able to tractably reason about the uncertainty through a range of useful inference queries. We empirically show how probabilistic circuits can be used as an augmented representation for structure learning methods, leading to improvement in both the quality of inferred structures and posterior uncertainty. Experimental results on conditional query answering further demonstrate the practical utility of the representational capacity of TRUST.},
	language = {en},
	author = {Wang, Benjie and Wicker, Matthew and Kwiatkowska, Marta},
	pages = {20},
	file = {Wang et al. - Tractable Uncertainty for Structure Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/LUUSXXGV/Wang et al. - Tractable Uncertainty for Structure Learning.pdf:application/pdf},
}

@article{chen_causal_2022,
	title = {Causal {Discovery} in {Linear} {Non}-{Gaussian} {Acyclic} {Model} {With} {Multiple} {Latent} {Confounders}},
	volume = {33},
	issn = {2162-237X, 2162-2388},
	url = {https://ieeexplore.ieee.org/document/9317707/},
	doi = {10.1109/TNNLS.2020.3045812},
	abstract = {Causal discovery from observational data is a fundamental problem in science. Though the linear non-Gaussian acyclic model (LiNGAM) has shown promising results in various applications, it still faces the following challenges in the data with multiple latent confounders: 1) how to detect the latent confounders and 2) how to uncover the causal relations among observed and latent variables. To address these two challenges, we propose a hybrid causal discovery method for the LiNGAM with multiple latent confounders (MLCLiNGAM). First, we utilize the constraint-based method to learn the causal skeleton. Second, we identify the causal directions, by conducting regression and independence tests on the adjacent pairs in the causal skeleton. Third, we detect the latent confounders with the help of the maximal clique patterns raised by the latent confounders and reconstruct the causal structure with latent variables. Theoretical results show the correctness and efﬁciency of the algorithms. We conduct extensive experiments on synthetic and real data, which illustrates the efﬁciency and effectiveness of the proposed algorithms.},
	language = {en},
	number = {7},
	urldate = {2022-11-08},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	author = {Chen, Wei and Cai, Ruichu and Zhang, Kun and Hao, Zhifeng},
	month = jul,
	year = {2022},
	pages = {2816--2827},
	file = {Chen et al. - 2022 - Causal Discovery in Linear Non-Gaussian Acyclic Mo.pdf:/Users/jacquelinemaasch/Zotero/storage/455BPIYE/Chen et al. - 2022 - Causal Discovery in Linear Non-Gaussian Acyclic Mo.pdf:application/pdf},
}

@misc{forre_constraint-based_2018,
	title = {Constraint-based {Causal} {Discovery} for {Non}-{Linear} {Structural} {Causal} {Models} with {Cycles} and {Latent} {Confounders}},
	url = {http://arxiv.org/abs/1807.03024},
	abstract = {We address the problem of causal discovery from data, making use of the recently proposed causal modeling framework of modular structural causal models (mSCM) to handle cycles, latent confounders and non-linearities. We introduce σ-connection graphs (σ-CG), a new class of mixed graphs (containing undirected, bidirected and directed edges) with additional structure, and extend the concept of σ-separation, the appropriate generalization of the well-known notion of d-separation in this setting, to apply to σ-CGs. We prove the closedness of σ-separation under marginalisation and conditioning and exploit this to implement a test of σ-separation on a σ-CG. This then leads us to the ﬁrst causal discovery algorithm that can handle non-linear functional relations, latent confounders, cyclic causal relationships, and data from different (stochastic) perfect interventions. As a proof of concept, we show on synthetic data how well the algorithm recovers features of the causal graph of modular structural causal models.},
	language = {en},
	urldate = {2022-11-08},
	publisher = {arXiv},
	author = {Forré, Patrick and Mooij, Joris M.},
	month = jul,
	year = {2018},
	note = {arXiv:1807.03024 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Forré and Mooij - 2018 - Constraint-based Causal Discovery for Non-Linear S.pdf:/Users/jacquelinemaasch/Zotero/storage/XM3XUYCF/Forré and Mooij - 2018 - Constraint-based Causal Discovery for Non-Linear S.pdf:application/pdf},
}

@article{zhang_completeness_2008,
	title = {On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias},
	volume = {172},
	issn = {00043702},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0004370208001008},
	doi = {10.1016/j.artint.2008.08.001},
	abstract = {Causal discovery becomes especially challenging when the possibility of latent confounding and/or selection bias is not assumed away. For this task, ancestral graph models are particularly useful in that they can represent the presence of latent confounding and selection effect, without explicitly invoking unobserved variables. Based on the machinery of ancestral graphs, there is a provably sound causal discovery algorithm, known as the FCI algorithm, that allows the possibility of latent confounders and selection bias. However, the orientation rules used in the algorithm are not complete. In this paper, we provide additional orientation rules, augmented by which the FCI algorithm is shown to be complete, in the sense that it can, under standard assumptions, discover all aspects of the causal structure that are uniquely determined by facts of probabilistic dependence and independence. The result is useful for developing any causal discovery and reasoning system based on ancestral graph models.},
	language = {en},
	number = {16-17},
	urldate = {2022-11-08},
	journal = {Artificial Intelligence},
	author = {Zhang, Jiji},
	month = nov,
	year = {2008},
	pages = {1873--1896},
	file = {Zhang - 2008 - On the completeness of orientation rules for causa.pdf:/Users/jacquelinemaasch/Zotero/storage/ALCRAPUV/Zhang - 2008 - On the completeness of orientation rules for causa.pdf:application/pdf},
}

@inproceedings{maeda_rcd_2020,
	title = {{RCD}: {Repetitive} causal discovery of linear non-{Gaussian} acyclic models with latent confounders},
	abstract = {Causal discovery from data aﬀected by latent confounders is an important and diﬃcult challenge. Causal functional model-based approaches have not been used to present variables whose relationships are aﬀected by latent confounders, while some constraintbased methods can present them. This paper proposes a causal functional modelbased method called repetitive causal discovery (RCD) to discover the causal structure of observed variables aﬀected by latent confounders. RCD repeats inferring the causal directions between a small number of observed variables and determines whether the relationships are aﬀected by latent confounders. RCD ﬁnally produces a causal graph where a bi-directed arrow indicates the pair of variables that have the same latent confounders, and a directed arrow indicates the causal direction of a pair of variables that are not aﬀected by the same latent confounder. The results of experimental validation using simulated data and real-world data conﬁrmed that RCD is eﬀective in identifying latent confounders and causal directions between observed variables.},
	booktitle = {Proceedings of the 23rd {International} {Conference} on {Artificial} {Intelligence} and {Statistics} ({AISTATS})},
	author = {Maeda, Takashi Nicholas and Shimizu, Shohei},
	year = {2020},
	file = {Maeda and Shimizu - RCD Repetitive causal discovery of linear non-Gau.pdf:/Users/jacquelinemaasch/Zotero/storage/6SYQTEQH/Maeda and Shimizu - RCD Repetitive causal discovery of linear non-Gau.pdf:application/pdf},
}

@article{raghu_comparison_2018,
	title = {Comparison of strategies for scalable causal discovery of latent variable models from mixed data},
	volume = {6},
	issn = {2364-415X, 2364-4168},
	url = {http://link.springer.com/10.1007/s41060-018-0104-3},
	doi = {10.1007/s41060-018-0104-3},
	abstract = {Modern technologies allow large, complex biomedical datasets to be collected from patient cohorts. These datasets are comprised of both continuous and categorical data (“Mixed Data”), and essential variables may be unobserved in this data due to the complex nature of biomedical phenomena. Causal inference algorithms can identify important relationships from biomedical data; however, handling the challenges of causal inference over mixed data with unmeasured confounders in a scalable way is still an open problem. Despite recent advances into causal discovery strategies that could potentially handle these challenges; individually, no study currently exists that comprehensively compares these approaches in this setting. In this paper, we present a comparative study that addresses this problem by comparing the accuracy and efﬁciency of different strategies in large, mixed datasets with latent confounders. We experiment with two extensions of the Fast Causal Inference algorithm: a maximum probability search procedure we recently developed to identify causal orientations more accurately, and a strategy which quickly eliminates unlikely adjacencies in order to achieve scalability to high-dimensional data. We demonstrate that these methods signiﬁcantly outperform the state of the art in the ﬁeld by achieving both accurate edge orientations and tractable running time in simulation experiments on datasets with up to 500 variables. Finally, we demonstrate the usability of the best performing approach on real data by applying it to a biomedical dataset of HIV-infected individuals.},
	language = {en},
	number = {1},
	urldate = {2022-11-08},
	journal = {International Journal of Data Science and Analytics},
	author = {Raghu, Vineet K. and Ramsey, Joseph D. and Morris, Alison and Manatakis, Dimitrios V. and Sprites, Peter and Chrysanthis, Panos K. and Glymour, Clark and Benos, Panayiotis V.},
	month = aug,
	year = {2018},
	pages = {33--45},
	file = {Raghu et al. - 2018 - Comparison of strategies for scalable causal disco.pdf:/Users/jacquelinemaasch/Zotero/storage/ADYNP3MJ/Raghu et al. - 2018 - Comparison of strategies for scalable causal disco.pdf:application/pdf},
}

@inproceedings{zeng_causal_2021,
	address = {Montreal, Canada},
	title = {Causal {Discovery} with {Multi}-{Domain} {LiNGAM} for {Latent} {Factors}},
	isbn = {978-0-9992411-9-6},
	url = {https://www.ijcai.org/proceedings/2021/289},
	doi = {10.24963/ijcai.2021/289},
	abstract = {Discovering causal structures among latent factors from observed data is particularly signiﬁcant yet challenging problem. Despite some eﬀorts for this problem, existing methods focus on the single-domain data only. In this paper, we propose Multi-Domain Linear NonGaussian Acyclic Models for LAtent Factors (MD-LiNA), where the causal structures from diﬀerent domains may be diﬀerent, but they have a shared causal structure among latent factors of interest. The model enriches the causal representation for multi-domain data. We propose an integrated two-phase algorithm to estimate the model. In particular, we ﬁrst locate the latent factors and estimate the factor loading matrix. Then to uncover the shared causal structure among latent factors of interest, we derive a score function based on the characterization of independence relations between external inﬂuences and the dependence relations between multi-domain latent factors and latent factors of interest. Experimental results on synthetic data demonstrate the eﬃcacy of our approach.},
	language = {en},
	urldate = {2022-11-08},
	booktitle = {Proceedings of the {Thirtieth} {International} {Joint} {Conference} on {Artificial} {Intelligence}},
	publisher = {International Joint Conferences on Artificial Intelligence Organization},
	author = {Zeng, Yan and Shimizu, Shohei and Cai, Ruichu and Xie, Feng and Yamamoto, Michio and Hao, Zhifeng},
	month = aug,
	year = {2021},
	pages = {2097--2103},
	file = {Zeng et al. - 2021 - Causal Discovery with Multi-Domain LiNGAM for Late.pdf:/Users/jacquelinemaasch/Zotero/storage/NWL2NII6/Zeng et al. - 2021 - Causal Discovery with Multi-Domain LiNGAM for Late.pdf:application/pdf},
}

@article{gerhardus_high-recall_nodate,
	title = {High-recall causal discovery for autocorrelated time series with latent confounders},
	abstract = {We present a new method for linear and nonlinear, lagged and contemporaneous constraint-based causal discovery from observational time series in the presence of latent confounders. We show that existing causal discovery methods such as FCI and variants suffer from low recall in the autocorrelated time series case and identify low effect size of conditional independence tests as the main reason. Information-theoretical arguments show that effect size can often be increased if causal parents are included in the conditioning sets. To identify parents early on, we suggest an iterative procedure that utilizes novel orientation rules to determine ancestral relationships already during the edge removal phase. We prove that the method is order-independent, and sound and complete in the oracle case. Extensive simulation studies for different numbers of variables, time lags, sample sizes, and further cases demonstrate that our method indeed achieves much higher recall than existing methods for the case of autocorrelated continuous variables while keeping false positives at the desired level. This performance gain grows with stronger autocorrelation. At github.com/jakobrunge/tigramite we provide Python code for all methods involved in the simulation studies.},
	language = {en},
	author = {Gerhardus, Andreas and Runge, Jakob},
	pages = {11},
	file = {Gerhardus and Runge - High-recall causal discovery for autocorrelated ti.pdf:/Users/jacquelinemaasch/Zotero/storage/N5MKHCFD/Gerhardus and Runge - High-recall causal discovery for autocorrelated ti.pdf:application/pdf},
}

@article{peters2014identifiability,
  author={Peters, J. and Bühlmann, P.},
  title={Identifiability of Gaussian structural equation models with equal error variances},
  journal={Biometrika},
  volume={101},
  number={1},
  pages={219--228},
  year={2014},
  month={March},
  doi={10.1093/biomet/ast043}
}


@article{spirtes_causal_2016,
	title = {Causal discovery and inference: concepts and recent methodological advances},
	volume = {3},
	issn = {2196-0089},
	shorttitle = {Causal discovery and inference},
	doi = {10.1186/s40535-016-0018-x},
	abstract = {This paper aims to give a broad coverage of central concepts and principles involved in automated causal inference and emerging approaches to causal discovery from i.i.d data and from time series. After reviewing concepts including manipulations, causal models, sample predictive modeling, causal predictive modeling, and structural equation models, we present the constraint-based approach to causal discovery, which relies on the conditional independence relationships in the data, and discuss the assumptions underlying its validity. We then focus on causal discovery based on structural equations models, in which a key issue is the identifiability of the causal structure implied by appropriately defined structural equation models: in the two-variable case, under what conditions (and why) is the causal direction between the two variables identifiable? We show that the independence between the error term and causes, together with appropriate structural constraints on the structural equation, makes it possible. Next, we report some recent advances in causal discovery from time series. Assuming that the causal relations are linear with nonGaussian noise, we mention two problems which are traditionally difficult to solve, namely causal discovery from subsampled data and that in the presence of confounding time series. Finally, we list a number of open questions in the field of causal discovery and inference.},
	language = {en},
	number = {1},
	urldate = {2022-11-08},
	journal = {Applied Informatics},
	author = {Spirtes, Peter and Zhang, Kun},
	month = dec,
	year = {2016},
	pages = {3},
	file = {Spirtes and Zhang - 2016 - Causal discovery and inference concepts and recen.pdf:/Users/jacquelinemaasch/Zotero/storage/DEK95AL2/Spirtes and Zhang - 2016 - Causal discovery and inference concepts and recen.pdf:application/pdf},
}

@misc{maeda_discovery_2021,
	title = {Discovery of {Causal} {Additive} {Models} in the {Presence} of {Unobserved} {Variables}},
	url = {http://arxiv.org/abs/2106.02234},
	abstract = {Causal discovery from data affected by unobserved variables is an important but difﬁcult problem to solve. The effects that unobserved variables have on the relationships between observed variables are more complex in nonlinear cases than in linear cases. In this study, we focus on causal additive models in the presence of unobserved variables. Causal additive models exhibit structural equations that are additive in the variables and error terms. We take into account the presence of not only unobserved common causes but also unobserved intermediate variables. Our theoretical results show that, when the causal relationships are nonlinear and there are unobserved variables, it is not possible to identify all the causal relationships between observed variables through regression and independence tests. However, our theoretical results also show that it is possible to avoid incorrect inferences. We propose a method to identify all the causal relationships that are theoretically possible to identify without being biased by unobserved variables. The empirical results using artiﬁcial data and simulated functional magnetic resonance imaging (fMRI) data show that our method effectively infers causal structures in the presence of unobserved variables.},
	language = {en},
	urldate = {2022-11-08},
	publisher = {arXiv},
	author = {Maeda, Takashi Nicholas and Shimizu, Shohei},
	month = jun,
	year = {2021},
	note = {arXiv:2106.02234 [cs]},
	keywords = {Computer Science - Machine Learning},
	file = {Maeda and Shimizu - 2021 - Discovery of Causal Additive Models in the Presenc.pdf:/Users/jacquelinemaasch/Zotero/storage/4J826EXQ/Maeda and Shimizu - 2021 - Discovery of Causal Additive Models in the Presenc.pdf:application/pdf},
}

@article{hadamard_bio,
  title={An extension of the Walsh-Hadamard transform to calculate and model epistasis in genetic landscapes of arbitrary shape and complexity},
  author={Faure, Andre J and Lehner, Ben and Mir{\'o} Pina, Ver{\'o}nica and Serrano Colome, Claudia and Weghorn, Donate},
  journal={PLOS Computational Biology},
  volume={20},
  number={5},
  pages={e1012132},
  year={2024},
  publisher={Public Library of Science San Francisco, CA USA}
}

@article{xie_generalized_2020,
	title = {Generalized {Independent} {Noise} {Condition} for {Estimating} {Latent} {Variable} {Causal} {Graphs}},
	abstract = {Causal discovery aims to recover causal structures or models underlying the observed data. Despite its success in certain domains, most existing methods focus on causal relations between observed variables, while in many scenarios the observed ones may not be the underlying causal variables (e.g., image pixels), but are generated by latent causal variables or confounders that are causally related. To this end, in this paper, we consider Linear, Non-Gaussian Latent variable Models (LiNGLaMs), in which latent confounders are also causally related, and propose a Generalized Independent Noise (GIN) condition to estimate such latent variable graphs. Speciﬁcally, for two observed random vectors Y and Z, GIN holds if and only if ω⊺Y and Z are statistically independent, where ω is a parameter vector characterized from the cross-covariance between Y and Z. From the graphical view, roughly speaking, GIN implies that causally earlier latent common causes of variables in Y d-separate Y from Z. Interestingly, we ﬁnd that the independent noise condition, i.e., if there is no confounder, causes are independent from the error of regressing the effect on the causes, can be seen as a special case of GIN. Moreover, we show that GIN helps locate latent variables and identify their causal structure, including causal directions. We further develop a recursive learning algorithm to achieve these goals. Experimental results on synthetic and real-world data demonstrate the effectiveness of our method.},
	language = {en},
	journal = {34th Conference on Neural Information Processing Systems},
	author = {Xie, Feng and Cai, Ruichu and Huang, Biwei and Glymour, Clark and Hao, Zhifeng and Zhang, Kun},
	year = {2020},
	pages = {12},
	file = {Xie et al. - Generalized Independent Noise Condition for Estima.pdf:/Users/jacquelinemaasch/Zotero/storage/JSAERXCW/Xie et al. - Generalized Independent Noise Condition for Estima.pdf:application/pdf},
}

@misc{geffner_deep_2022,
	title = {Deep {End}-to-end {Causal} {Inference}},
	url = {http://arxiv.org/abs/2202.02195},
	abstract = {Causal inference is essential for data-driven decision making across domains such as business engagement, medical treatment and policy making. However, research on causal discovery has evolved separately from inference methods, preventing straight-forward combination of methods from both ﬁelds. In this work, we develop Deep End-to-end Causal Inference (DECI), a single ﬂow-based non-linear additive noise model that takes in observational data and can perform both causal discovery and inference, including conditional average treatment effect (CATE) estimation. We provide a theoretical guarantee that DECI can recover the ground truth causal graph under standard causal discovery assumptions. Motivated by application impact, we extend this model to heterogeneous, mixed-type data with missing values, allowing for both continuous and discrete treatment decisions. Our results show the competitive performance of DECI when compared to relevant baselines for both causal discovery and (C)ATE estimation in over a thousand experiments on both synthetic datasets and causal machine learning benchmarks across data-types and levels of missingness.},
	language = {en},
	urldate = {2022-11-23},
	publisher = {arXiv},
	author = {Geffner, Tomas and Antoran, Javier and Foster, Adam and Gong, Wenbo and Ma, Chao and Kiciman, Emre and Sharma, Amit and Lamb, Angus and Kukla, Martin and Pawlowski, Nick and Allamanis, Miltiadis and Zhang, Cheng},
	month = jun,
	year = {2022},
	note = {arXiv:2202.02195 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Geffner et al. - 2022 - Deep End-to-end Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/LMAHHWBS/Geffner et al. - 2022 - Deep End-to-end Causal Inference.pdf:application/pdf},
}

@inproceedings{peter_spirtes_causal_1995,
	title = {Causal inference in the presence of latent variables and selection bias},
	doi = {https://doi.org/10.1007/s13398-014-0173-7.2},
	booktitle = {Proceedings of the {Eleventh} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {{Peter Spirtes} and {Christopher Meek} and {Thomas Richardson}},
	year = {1995},
	pages = {499--506},
	file = {Peter Spirtes et al. - 1995 - Causal inference in the presence of latent variabl.pdf:/Users/jacquelinemaasch/Zotero/storage/JABGL5YB/Peter Spirtes et al. - 1995 - Causal inference in the presence of latent variabl.pdf:application/pdf},
}

@inproceedings{maeda_causal_2021,
	title = {Causal {Additive} {Models} with {Unobserved} {Variables}},
	abstract = {Causal discovery from data affected by unobserved variables is an important but difﬁcult problem to solve. The effects that unobserved variables have on the relationships between observed variables are more complex in nonlinear cases than in linear cases. In this study, we focus on causal additive models in the presence of unobserved variables. Causal additive models exhibit structural equations that are additive in the variables and error terms. We take into account the presence of not only unobserved common causes but also unobserved intermediate variables. Our theoretical results show that, when the causal relationships are nonlinear and there are unobserved variables, it is not possible to identify all the causal relationships between observed variables through regression and independence tests. However, our theoretical results also show that it is possible to avoid incorrect inferences. We propose a method to identify all the causal relationships that are theoretically possible to identify without being biased by unobserved variables. The empirical results using artiﬁcial data and simulated functional magnetic resonance imaging (fMRI) data show that our method effectively infers causal structures in the presence of unobserved variables.},
	language = {en},
	booktitle = {Proceedings of the {Thirty}-{Seventh} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Maeda, Takashi Nicholas and Shimizu, Shohei},
	year = {2021},
	pages = {10},
	file = {Maeda and Shimizu - 2021 - Causal Additive Models with Unobserved Variables.pdf:/Users/jacquelinemaasch/Zotero/storage/CVYUBE7L/Maeda and Shimizu - 2021 - Causal Additive Models with Unobserved Variables.pdf:application/pdf},
}

@article{zhang_kernel-based_2011,
	title = {Kernel-based {Conditional} {Independence} {Test} and {Application} in {Causal} {Discovery}},
	abstract = {Conditional independence testing is an important problem, especially in Bayesian network learning and causal discovery. Due to the curse of dimensionality, testing for conditional independence of continuous variables is particularly challenging. We propose a Kernel-based Conditional Independence test (KCI-test), by constructing an appropriate test statistic and deriving its asymptotic distribution under the null hypothesis of conditional independence. The proposed method is computationally eﬃcient and easy to implement. Experimental results show that it outperforms other methods, especially when the conditioning set is large or the sample size is not very large, in which case other methods encounter diﬃculties.},
	language = {en},
	journal = {Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence},
	author = {Zhang, Kun and Peters, Jonas and Janzing, Dominik and Scholkopf, Bernhard},
	year = {2011},
	file = {Zhang et al. - 2011 - Kernel-based Conditional Independence Test and App.pdf:/Users/jacquelinemaasch/Zotero/storage/VC2QAMGL/Zhang et al. - 2011 - Kernel-based Conditional Independence Test and App.pdf:application/pdf},
}

@inproceedings{maasch2024local,
      title={Local Discovery by Partitioning: Polynomial-Time Causal Discovery Around Exposure-Outcome Pairs},
      author={Jacqueline Maasch and Weishen Pan and Shantanu Gupta and Volodymyr Kuleshov and Kyra Gan and Fei Wang},
      year={2024},
      booktitle = {Proceedings of the 40th Conference on Uncertainy in Artificial Intelligence},
      eprint={2310.17816},
      doi = {https://doi.org/10.48550/arXiv.2310.17816},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@article{cheng_toward_2023,
	title = {Toward {Unique} and {Unbiased} {Causal} {Effect} {Estimation} {From} {Data} {With} {Hidden} {Variables}},
	volume = {34},
	issn = {2162-237X, 2162-2388},
	url = {https://ieeexplore.ieee.org/document/9674196/},
	doi = {10.1109/TNNLS.2021.3133337},
	language = {en},
	number = {9},
	urldate = {2023-11-29},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	author = {Cheng, Debo and Li, Jiuyong and Liu, Lin and Yu, Kui and Duy Le, Thuc and Liu, Jixue},
	month = sep,
	year = {2023},
	pages = {6108--6120},
	file = {Cheng et al. - 2023 - Toward Unique and Unbiased Causal Effect Estimatio.pdf:/Users/jacquelinemaasch/Zotero/storage/68BG62P4/Cheng et al. - 2023 - Toward Unique and Unbiased Causal Effect Estimatio.pdf:application/pdf},
}

@article{mooij2016distinguishing,
  title={Distinguishing cause from effect using observational data: methods and benchmarks},
  author={Mooij, Joris M and Peters, Jonas and Janzing, Dominik and Zscheischler, Jakob and Sch{\"o}lkopf, Bernhard},
  journal={Journal of Machine Learning Research},
  volume={17},
  number={32},
  pages={1--102},
  year={2016}
}

@article{hastie1987generalized,
  title={Generalized additive models: some applications},
  author={Hastie, Trevor and Tibshirani, Robert},
  journal={Journal of the American Statistical Association},
  volume={82},
  number={398},
  pages={371--386},
  year={1987},
  publisher={Taylor \& Francis}
}

@article{shah2024front,
  title={Front-door Adjustment Beyond Markov Equivalence with Limited Graph Knowledge},
  author={Shah, Abhin and Shanmugam, Karthikeyan and Kocaoglu, Murat},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}


@book{peter_spirtes_causation_2000,
	address = {New York, NY},
	series = {Lecture {Notes} in {Statistics}},
	title = {Causation, {Prediction}, and {Search}},
	volume = {81},
	isbn = {978-1-4612-7650-0 978-1-4612-2748-9},
	language = {en},
	urldate = {2022-11-26},
	publisher = {Springer New York},
	author = {Spirtes, Peter and Glymour, Clark and Scheines, Richard},
	year = {2000},
	doi = {10.1007/978-1-4612-2748-9},
	file = {Peter Spirtes et al. - 2000 - Causation, Prediction, and Search.pdf:/Users/jacquelinemaasch/Zotero/storage/TJGTUSK2/Peter Spirtes et al. - 2000 - Causation, Prediction, and Search.pdf:application/pdf},
}

@book{pearl_causal_2016,
	address = {Chichester, West Sussex},
	title = {Causal inference in statistics: a primer},
	isbn = {978-1-119-18684-7},
	shorttitle = {Causal inference in statistics},
	language = {en},
	publisher = {Wiley},
	author = {Pearl, Judea and Glymour, Madelyn and Jewell, Nicholas P.},
	year = {2016},
	keywords = {Causation, Mathematical statistics, Probabilities},
	file = {Pearl et al. - 2016 - Causal inference in statistics a primer.pdf:/Users/jacquelinemaasch/Zotero/storage/YZPFXH6G/Pearl et al. - 2016 - Causal inference in statistics a primer.pdf:application/pdf},
}

@article{pearl_causal_2009,
	title = {Causal inference in statistics: {An} overview},
	volume = {3},
	issn = {1935-7516},
	shorttitle = {Causal inference in statistics},
	doi = {10.1214/09-SS057},
	abstract = {This review presents empirical researchers with recent advances in causal inference, and stresses the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underly all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and uniﬁes other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the eﬀects of potential interventions, (also called “causal eﬀects” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret,” “attribution” or “causes of eﬀects”) and (3) queries about direct and indirect eﬀects (also known as “mediation”). Finally, the paper deﬁnes the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both.},
	language = {en},
	urldate = {2022-11-28},
	journal = {Statistics Surveys},
	author = {Pearl, Judea},
	month = jan,
	year = {2009},
	file = {Pearl - 2009 - Causal inference in statistics An overview.pdf:/Users/jacquelinemaasch/Zotero/storage/WZ7CWN4V/Pearl - 2009 - Causal inference in statistics An overview.pdf:application/pdf},
}

@article{weinberger_comparing_nodate,
	title = {Comparing {Rubin} and {Pearl}’s {Causal} {Modeling} {Frameworks}: {A} {Commentary} on {Markus} (2021)},
	abstract = {Markus (2021) argues that the causal modeling frameworks of Pearl and Rubin are not “strongly equivalent”, in the sense of saying “the same thing in diﬀerent ways”. Here I rebut Markus’ arguments against strong equivalence. The diﬀerences between the frameworks are best illuminated not by appeal to their causal semantics, but rather reﬂect pragmatic modeling choices.},
	language = {en},
	author = {Weinberger, Naftali},
	pages = {9},
	file = {Weinberger - Comparing Rubin and Pearl’s Causal Modeling Framew.pdf:/Users/jacquelinemaasch/Zotero/storage/29J2AE99/Weinberger - Comparing Rubin and Pearl’s Causal Modeling Framew.pdf:application/pdf},
}

@inproceedings{huang_generalized_2018,
	address = {London United Kingdom},
	title = {Generalized {Score} {Functions} for {Causal} {Discovery}},
	isbn = {978-1-4503-5552-0},
	url = {https://dl.acm.org/doi/10.1145/3219819.3220104},
	doi = {10.1145/3219819.3220104},
	abstract = {Discovery of causal relationships from observational data is a fundamental problem. Roughly speaking, there are two types of methods for causal discovery, constraint-based ones and score-based ones. Score-based methods avoid the multiple testing problem and enjoy certain advantages compared to constraint-based ones. However, most of them need strong assumptions on the functional forms of causal mechanisms, as well as on data distributions, which limit their applicability. In practice the precise information of the underlying model class is usually unknown. If the above assumptions are violated, both spurious and missing edges may result. In this paper, we introduce generalized score functions for causal discovery based on the characterization of general (conditional) independence relationships between random variables, without assuming particular model classes. In particular, we exploit regression in RKHS to capture the dependence in a nonparametric way. The resulting causal discovery approach produces asymptotically correct results in rather general cases, which may have nonlinear causal mechanisms, a wide class of data distributions, mixed continuous and discrete data, and multidimensional variables. Experimental results on both synthetic and real-world data demonstrate the efficacy of our proposed approach.},
	language = {en},
	urldate = {2022-11-29},
	booktitle = {Proceedings of the 24th {ACM} {SIGKDD} {International} {Conference} on {Knowledge} {Discovery} \& {Data} {Mining}},
	publisher = {ACM},
	author = {Huang, Biwei and Zhang, Kun and Lin, Yizhu and Schölkopf, Bernhard and Glymour, Clark},
	month = jul,
	year = {2018},
	pages = {1551--1560},
	file = {Huang et al. - 2018 - Generalized Score Functions for Causal Discovery.pdf:/Users/jacquelinemaasch/Zotero/storage/4W2J8TUN/Huang et al. - 2018 - Generalized Score Functions for Causal Discovery.pdf:application/pdf},
}

@inproceedings{ramsey_tetrad_2018,
	title = {{TETRAD} - {A} toolbox for causal discovery},
	abstract = {Climate and Earth research seeks to identify causal relationships without the advantage of experimental controls. Algorithmic methods for that purpose have a long history and have developed rapidly in the last quarter century, so that new methods appear almost monthly. Unsurprisingly, these products vary in accuracy, informativeness, quality of implementation, and necessary assumptions. Researchers need a guide and implementation of well-vetted causal search methods and a means to test and compare methods on real and simulated data. We review the TETRAD suite of programs for these purposes, available from the Pittsburgh/Carnegie Mellon Center for Causal (http://www.phil.cmu.edu/tetrad/).},
	language = {en},
	booktitle = {8th {International} {Workshop} on {Climate} {Informatics}},
	author = {Ramsey, Joseph D and Zhang, Kun and Glymour, Madelyn and Romero, Ruben Sanchez and Huang, Biwei and Samarasinghe, Savini and Barnes, Elizabeth A and Glymour, Clark},
	year = {2018},
	file = {Ramsey et al. - 2018 - TETRAD - A toolbox for causal discovery.pdf:/Users/jacquelinemaasch/Zotero/storage/LHQA2QCB/Ramsey et al. - 2018 - TETRAD - A toolbox for causal discovery.pdf:application/pdf},
}

@article{rabaey_neural_nodate,
	title = {Neural {Bayesian} {Network} {Understudy}},
	abstract = {Bayesian Networks may be appealing for clinical decision-making due to their inclusion of causal knowledge, but their practical adoption remains limited as a result of their inability to deal with unstructured data. While neural networks do not have this limitation, they are not interpretable and are inherently unable to deal with causal structure in the input space. Our goal is to build neural networks that combine the advantages of both approaches. Motivated by the perspective to inject causal knowledge while training such neural networks, this work presents initial steps in that direction. We demonstrate how a neural network can be trained to output conditional probabilities, providing approximately the same functionality as a Bayesian Network. Additionally, we propose two training strategies that allow encoding the independence relations inferred from a given causal structure into the neural network. We present initial results in a proof-of-concept setting, showing that the neural model acts as an understudy to its Bayesian Network counterpart, approximating its probabilistic and causal properties.},
	language = {en},
	author = {Rabaey, Paloma},
	pages = {12},
	file = {Rabaey - Neural Bayesian Network Understudy.pdf:/Users/jacquelinemaasch/Zotero/storage/ZYSJRC65/Rabaey - Neural Bayesian Network Understudy.pdf:application/pdf},
}

@article{poudel_contrastive_nodate,
	title = {Contrastive {Unsupervised} {Learning} of {World} {Model} with {Invariant} {Causal} {Features}},
	language = {en},
	author = {Poudel, Rudra P K and Pandya, Harit and Cipolla, Roberto},
	pages = {12},
	file = {Poudel et al. - Contrastive Unsupervised Learning of World Model w.pdf:/Users/jacquelinemaasch/Zotero/storage/DTDB3D8Y/Poudel et al. - Contrastive Unsupervised Learning of World Model w.pdf:application/pdf},
}

@article{cai_improving_nodate,
	title = {Improving the {Efficiency} of the {PC} {Algorithm} by {Using} {Model}-{Based} {Conditional} {Independence} {Tests}},
	abstract = {Learning causal structure is useful in many areas of artificial intelligence, including planning, robotics, and explanation. Constraint-based structure learning algorithms such as PC use conditional independence (CI) tests to infer causal structure. Traditionally, constraint-based algorithms perform CI tests with a preference for smaller-sized conditioning sets, partially because the statistical power of conventional CI tests declines rapidly as the size of the conditioning set increases. However, many modern conditional independence tests are model-based, and these tests use well-regularized models that maintain statistical power even with very large conditioning sets. This suggests an intriguing new strategy for constraintbased algorithms which may result in a reduction of the total number of CI tests performed: Test variable pairs with large conditioning sets first, as a pre-processing step that finds some conditional independencies quickly, before moving on to the more conventional strategy that favors small conditioning sets. We propose such a pre-processing step for the PC algorithm which relies on performing CI tests on a few randomly selected large conditioning sets. We perform an empirical analysis on directed acyclic graphs (DAGs) that correspond to real-world systems and both empirical and theoretical analyses for Erdo˝s-Renyi DAGs. Our results show that Pre-Processing Plus PC (P3PC) performs far fewer CI tests than the original PC algorithm, between 0.5\% to 36\%, and often less than 10\%, of the CI tests that the PC algorithm alone performs. The efficiency gains are particularly significant for the DAGs corresponding to real-world systems.},
	language = {en},
	author = {Cai, Erica and McGregor, Andrew and Jensen, David},
	pages = {8},
	file = {Cai et al. - Improving the Efficiency of the PC Algorithm by Us.pdf:/Users/jacquelinemaasch/Zotero/storage/EJKZ56AD/Cai et al. - Improving the Efficiency of the PC Algorithm by Us.pdf:application/pdf},
}

@article{long_can_nodate,
	title = {Can {Large} {Language} {Models} {Build} {Causal} {Graphs}?},
	abstract = {Building causal graphs can be a laborious process. To ensure all relevant causal pathways have been captured, researchers often have to discuss with clinicians and experts while also reviewing extensive relevant medical literature. By encoding common and medical knowledge, large language models (LLMs) represent an opportunity to ease this process by automatically scoring edges (i.e., connections between two variables) in potential graphs. LLMs however have been shown to be brittle to the choice of probing words, context, and prompts that the user employs. In this work, we evaluate if LLMs can be a useful tool in complementing causal graph development.},
	language = {en},
	author = {Long, Stephanie and Schuster, Tibor and Piché, Alexandre},
	pages = {8},
	file = {Long et al. - Can Large Language Models Build Causal Graphs.pdf:/Users/jacquelinemaasch/Zotero/storage/TCE4MJ4J/Long et al. - Can Large Language Models Build Causal Graphs.pdf:application/pdf},
}

@article{gong_rhino_nodate,
	title = {Rhino: {Deep} {Causal} {Temporal} {Relationship} {Learning} with history-dependent noise},
	abstract = {Discovering causal relationships between different variables from time series data has been a long-standing challenge for many domains. For example, in stock markets, the announcement of acquisitions from leading companies may have immediate effects on stock prices and increased uncertainty of the future market due to this past action. This requires the model to take non-linear relationships, instantaneous effects and the past-action dependent uncertainty into account. We name the latter as history-dependent noise. However, previous works do not offer a solution addressing all these problems together. In this paper, we propose a structural equation model, called Rhino, which combines vector auto-regression, deep learning and variational inference to model non-linear relationships with instantaneous effects and flexible history-dependent noise. Theoretically, we prove the structural identifiability for a generalization of Rhino. Our empirical results from extensive synthetic experiments and a real-world benchmark demonstrate better discovery performance compared to relevant baselines, with ablation studies revealing its robustness when the Rhino is misspecified.},
	language = {en},
	author = {Gong, Wenbo and Jennings, Joel and Zhang, Cheng and Pawlowski, Nick},
	pages = {28},
	file = {Gong et al. - Rhino Deep Causal Temporal Relationship Learning .pdf:/Users/jacquelinemaasch/Zotero/storage/YU2XCALS/Gong et al. - Rhino Deep Causal Temporal Relationship Learning .pdf:application/pdf},
}

@article{ashman_causal_nodate,
	title = {Causal {Reasoning} in the {Presence} of {Latent} {Confounders} via {Neural} {ADMG} {Learning}},
	abstract = {Latent confounding has been a long-standing obstacle for causal reasoning from observational data. One popular approach is to model the data using acyclic directed mixed graphs (ADMGs), which describe ancestral relations between variables using directed and bidirected edges. However, existing methods using ADMGs are based on either linear functional assumptions or a discrete search that is complicated to use and lacks computational tractability for large datasets. In this work, we further extend the existing body of work and develop a novel gradientbased approach to learning an ADMG with non-linear functional relations from observational data. We first show that the presence of latent confounding is identifiable under the assumptions of bow-free ADMGs with non-linear additive noise models. With this insight, we propose a novel neural causal model based on autoregressive flows. This not only enables us to model complex causal relationships behind the data, but also estimate their functional relationships (hence treatment effects) simultaneously. We further validate our approach via experiments on both synthetic and real-world datasets, and demonstrate the competitive performance against relevant baselines.},
	language = {en},
	author = {Ashman, Matthew and Ma, Chao and Hilmkil, Agrin},
	pages = {18},
	file = {Ashman et al. - Causal Reasoning in the Presence of Latent Confoun.pdf:/Users/jacquelinemaasch/Zotero/storage/GGKSQZ4P/Ashman et al. - Causal Reasoning in the Presence of Latent Confoun.pdf:application/pdf},
}

@article{abdulaal_deep_nodate,
	title = {Deep {Structural} {Causal} {Modelling} of the {Clinical} and {Radiological} {Phenotype} of {Alzheimer}’s {Disease}},
	abstract = {Alzheimer’s disease (AD) has a poorly understood aetiology. Patients often have different rates and patterns of brain atrophy, and present at different stages along the natural history of their condition. This means that establishing the relationships between disease-related variables, and subsequently linking the clinical and radiological phenotypes of AD is difficult. Investigating this link is important because it could ultimately allow for a better understanding of the disease process, and this could enable tasks such as treatment effect estimates, disease progression modelling, and better precision medicine for AD patients. We extend a class of deep structural causal models (DSCMs) to the clinical and radiological phenotype of AD, and propose an aetiological model of relevant patient demographics, imaging and clinical biomarkers, and cognitive assessment/educational scores based on specific current hypotheses in the medical literature. The trained DSCM produces biologically plausible counterfactuals relating to the specified disease covariates, and reproduces ground-truth longitudinal changes in magnetic resonance images of AD. Such a model could enable the assessment of the effects of intervening on variables outside a randomized controlled trial setting. In addition, by being explicit about how causal relationships are encoded, the framework provides a principled approach to define and assess hypotheses of the aetiology of AD. Code to replicate the experiment can be found at: Counterfactual AD.},
	language = {en},
	author = {Abdulaal, Ahmed and Castro, Daniel C and Alexander, Daniel C},
	pages = {15},
	file = {Abdulaal et al. - Deep Structural Causal Modelling of the Clinical a.pdf:/Users/jacquelinemaasch/Zotero/storage/3J3M7D2P/Abdulaal et al. - Deep Structural Causal Modelling of the Clinical a.pdf:application/pdf},
}

@article{bennett_rethinking_nodate,
	title = {Rethinking {Neural} {Relational} {Inference} for {Granger} {Causal} {Discovery}},
	abstract = {Granger causal discovery aims to infer the underlying Granger causal relationships between pairs of variables in a multivariate time series system. Recent work has proposed using Neural Relational Inference (NRI) Kipf et al. [2018] – a latent graph inference model – for Granger causal discovery. However, the conditions under which NRI succeeds in recovering the true Granger causal graph remain unknown. In this work we show how the mean field approximation inherent in NRI has significant implications for its ability to recover the Granger causal structure in multivariate time series. We illustrate this point theoretically and experimentally using a linear vector autoregressive model – an important benchmark in economic and financial studies.},
	language = {en},
	author = {Bennett, Stefanos and Yu, Rose},
	pages = {9},
	file = {Bennett and Yu - Rethinking Neural Relational Inference for Granger.pdf:/Users/jacquelinemaasch/Zotero/storage/AZ6G4VXF/Bennett and Yu - Rethinking Neural Relational Inference for Granger.pdf:application/pdf},
}

@article{toth_active_nodate,
	title = {Active {Bayesian} {Causal} {Inference}},
	abstract = {Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference—other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.},
	language = {en},
	author = {Toth, Christian and Lorch, Lars and Knoll, Christian and Krause, Andreas and Pernkopf, Franz and Peharz, Robert},
	pages = {34},
	file = {Toth et al. - Active Bayesian Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/MIJ9IVB6/Toth et al. - Active Bayesian Causal Inference.pdf:application/pdf},
}

@article{dai_independence_nodate,
	title = {Independence {Testing}-{Based} {Approach} to {Causal} {Discovery} under {Measurement} {Error} and {Linear} {Non}-{Gaussian} {Models}},
	language = {en},
	author = {Dai, Haoyue and Spirtes, Peter and Zhang, Kun},
	pages = {36},
	file = {Dai et al. - Independence Testing-Based Approach to Causal Disc.pdf:/Users/jacquelinemaasch/Zotero/storage/Z3236NZG/Dai et al. - Independence Testing-Based Approach to Causal Disc.pdf:application/pdf},
}

@article{wang_sound_nodate,
	title = {Sound and {Complete} {Causal} {Identiﬁcation} with {Latent} {Variables} {Given} {Local} {Background} {Knowledge}},
	abstract = {Great efforts have been devoted to causal discovery from observational data, and it is well known that introducing some background knowledge attained from experiments or human expertise can be very helpful. However, it remains unknown that what causal relations are identiﬁable given background knowledge in the presence of latent confounders. In this paper, we solve the problem with sound and complete orientation rules when the background knowledge is given in a local form. Furthermore, based on the solution to the problem, this paper proposes a general active learning framework for causal discovery in the presence of latent confounders, with its effectiveness and efﬁciency validated by experiments.},
	language = {en},
	author = {Wang, Tian-Zuo and Qin, Tian and Zhou, Zhi-Hua},
	pages = {27},
	file = {Wang et al. - Sound and Complete Causal Identiﬁcation with Laten.pdf:/Users/jacquelinemaasch/Zotero/storage/PIMVX72H/Wang et al. - Sound and Complete Causal Identiﬁcation with Laten.pdf:application/pdf},
}

@article{gunther_conditional_nodate,
	title = {Conditional {Independence} {Testing} with {Heteroskedastic} {Data} and {Applications} to {Causal} {Discovery}},
	abstract = {Conditional independence (CI) testing is frequently used in data analysis and machine learning for various scientific fields and it forms the basis of constraint-based causal discovery. Oftentimes, CI testing relies on strong, rather unrealistic assumptions. One of these assumptions is homoskedasticity, in other words, a constant conditional variance is assumed. We frame heteroskedasticity in a structural causal model framework and present an adaptation of the partial correlation CI test that works well in the presence of heteroskedastic noise, given that expert knowledge about the heteroskedastic relationships is available. Further, we provide theoretical consistency results for the proposed CI test which carry over to causal discovery under certain assumptions. Numerical causal discovery experiments demonstrate that the adapted partial correlation CI test outperforms the standard test in the presence of heteroskedasticity and is on par for the homoskedastic case. Finally, we discuss the general challenges and limits as to how expert knowledge about heteroskedasticity can be accounted for in causal discovery.},
	language = {en},
	author = {Günther, Wiebke and Ninad, Urmi and Wahl, Jonas and Runge, Jakob},
	pages = {21},
	file = {Günther et al. - Conditional Independence Testing with Heteroskedas.pdf:/Users/jacquelinemaasch/Zotero/storage/G68CHNGA/Günther et al. - Conditional Independence Testing with Heteroskedas.pdf:application/pdf},
}

@article{yang_causal_nodate,
	title = {Causal {Discovery} in {Linear} {Latent} {Variable} {Models} {Subject} to {Measurement} {Error}},
	abstract = {We focus on causal discovery in the presence of measurement error in linear systems where the mixing matrix, i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables, is identiﬁed up to permutation and scaling of the columns. We demonstrate a somewhat surprising connection between this problem and causal discovery in the presence of unobserved parentless causes, in the sense that there is a mapping, given by the mixing matrix, between the underlying models to be inferred in these problems. Consequently, any identiﬁability result based on the mixing matrix for one model translates to an identiﬁability result for the other model. We characterize to what extent the causal models can be identiﬁed under a two-part faithfulness assumption. Under only the ﬁrst part of the assumption (corresponding to the conventional deﬁnition of faithfulness), the structure can be learned up to the causal ordering among an ordered grouping of the variables but not all the edges across the groups can be identiﬁed. We further show that if both parts of the faithfulness assumption are imposed, the structure can be learned up to a more reﬁned ordered grouping. As a result of this reﬁnement, for the latent variable model with unobserved parentless causes, the structure can be identiﬁed. Based on our theoretical results, we propose causal structure learning methods for both models, and evaluate their performance on synthetic data.},
	language = {en},
	author = {Yang, Yuqin and Ghassami, AmirEmad and Nafea, Mohamed and Kiyavash, Negar and Zhang, Kun and Shpitser, Ilya},
	pages = {33},
	file = {Yang et al. - Causal Discovery in Linear Latent Variable Models .pdf:/Users/jacquelinemaasch/Zotero/storage/PSGZ8RAK/Yang et al. - Causal Discovery in Linear Latent Variable Models .pdf:application/pdf},
}

@article{jeunen_disentangling_nodate,
	title = {Disentangling {Causal} {Effects} from {Sets} of {Interventions} in the {Presence} of {Unobserved} {Confounders}},
	abstract = {The ability to answer causal questions is crucial in many domains, as causal inference allows one to understand the impact of interventions. In many applications, only a single intervention is possible at a given time. However, in some important areas, multiple interventions are concurrently applied. Disentangling the effects of single interventions from jointly applied interventions is a challenging task—especially as simultaneously applied interventions can interact. This problem is made harder still by unobserved confounders, which influence both treatments and outcome. We address this challenge by aiming to learn the effect of a singleintervention from both observational data and sets of interventions. We prove that this is not generally possible, but provide identification proofs demonstrating that it can be achieved under non-linear continuous structural causal models with additive, multivariate Gaussian noise—even when unobserved confounders are present. Importantly, we show how to incorporate observed covariates and learn heterogeneous treatment effects. Based on the identifiability proofs, we provide an algorithm that learns the causal model parameters by pooling data from different regimes and jointly maximizing the combined likelihood. The effectiveness of our method is empirically demonstrated on both synthetic and real-world data.},
	language = {en},
	author = {Jeunen, Olivier and Gilligan-Lee, Ciarán M and Mehrotra, Rishabh and Lalmas, Mounia},
	pages = {15},
	file = {Jeunen et al. - Disentangling Causal Effects from Sets of Interven.pdf:/Users/jacquelinemaasch/Zotero/storage/BZNLPUK3/Jeunen et al. - Disentangling Causal Effects from Sets of Interven.pdf:application/pdf},
}

@article{lorch_amortized_nodate,
	title = {Amortized {Inference} for {Causal} {Structure} {Learning}},
	abstract = {Inferring causal structure poses a combinatorial search problem that typically involves evaluating structures with a score or independence test. The resulting search is costly, and designing suitable scores or tests that capture prior knowledge is difficult. In this work, we propose to amortize causal structure learning. Rather than searching over structures, we train a variational inference model to predict the causal structure from observational or interventional data. This allows us to bypass both the search over graphs and the hand-engineering of suitable score functions. Instead, our inference model acquires domain-specific inductive biases for causal discovery solely from data generated by a simulator. The architecture of our inference model emulates permutation invariances that are crucial for statistical efficiency in structure learning, which facilitates generalization to significantly larger problem instances than seen during training. On synthetic data and semisynthetic gene expression data, our models exhibit robust generalization capabilities when subject to substantial distribution shifts and significantly outperform existing algorithms, especially in the challenging genomics domain. Our code and models are publicly available at: https://github.com/larslorch/avici.},
	language = {en},
	author = {Lorch, Lars and Sussex, Scott and Rothfuss, Jonas and Krause, Andreas and Schölkopf, Bernhard},
	pages = {15},
	file = {Lorch et al. - Amortized Inference for Causal Structure Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/3YC7EUZ7/Lorch et al. - Amortized Inference for Causal Structure Learning.pdf:application/pdf},
}

@article{olko_trust_nodate,
	title = {Trust {Your} ∇: {Gradient}-based {Intervention} {Targeting} for {Causal} {Discovery}},
	abstract = {Inferring causal structure from data is a challenging task of fundamental importance in science. Observational data are often insufficient to identify a system’s causal structure uniquely. While conducting interventions (i.e., experiments) can improve the identifiability, such samples are usually challenging and expensive to obtain. Hence, experimental design approaches for causal discovery aim to minimize the number of interventions by estimating the most informative intervention target. In this work, we propose a novel Gradient-based Intervention Targeting method, abbreviated GIT, that ’trusts’ the gradient estimator of a gradient-based causal discovery framework to provide signals for the intervention acquisition function. We provide extensive experiments in simulated and real-world datasets and demonstrate that GIT performs on par with competitive baselines, surpassing them in the low-data regime.},
	language = {en},
	author = {Olko, Mateusz and Zaja, Michał and Nowak, Aleksandra},
	pages = {27},
	file = {Olko et al. - Trust Your ∇ Gradient-based Intervention Targetin.pdf:/Users/jacquelinemaasch/Zotero/storage/G4PCWZ7X/Olko et al. - Trust Your ∇ Gradient-based Intervention Targetin.pdf:application/pdf},
}

@article{gupta_local_nodate,
	title = {Local {Causal} {Discovery} for {Estimating} {Causal} {Effects}},
	abstract = {Even when the causal graph underlying our data is unknown, we can nevertheless use observational data to narrow down the possible values that an average treatment effect (ATE) can take by (1) identifying the graph up to a Markov equivalence class; and (2) estimating that ATE for each graph in the class. While the PC algorithm can identify this class under strong faithfulness assumptions, it can be computationally prohibitive. Fortunately, only the local graph structure around the treatment is required to identify an ATE, a fact exploited by local discovery algorithms to identify the possible values for an ATE more efficiently. In this paper, we introduce Local Discovery using Eager Collider Checks (LDECC), a new local discovery algorithm that leverages unshielded colliders to orient the treatment’s parents differently from existing methods. We show that there exist graphs where LDECC exponentially outperforms existing local discovery algorithms and vice versa. Moreover, we show that LDECC and existing algorithms rely on different faithfulness assumptions, leveraging this insight to test for and recover from certain faithfulness violations.},
	language = {en},
	author = {Gupta, Shantanu and Lipton, Zachary C and Childers, David},
	pages = {16},
	file = {Gupta et al. - Local Causal Discovery for Estimating Causal Effec.pdf:/Users/jacquelinemaasch/Zotero/storage/IWLEIB45/Gupta et al. - Local Causal Discovery for Estimating Causal Effec.pdf:application/pdf},
}

@article{deshpande_deep_nodate,
	title = {Deep {Multi}-{Modal} {Structural} {Equations} {For} {Causal} {Effect} {Estimation} {With} {Unstructured} {Proxies}},
	abstract = {Estimating the effect of an intervention from observational data while accounting for confounding variables is a key task in causal inference. Oftentimes, the confounders are unobserved, but we have access to large amounts of additional unstructured data (images, text) that contain valuable proxy signal about the missing confounders. This paper argues that leveraging this unstructured data can greatly improve the accuracy of causal effect estimation. Specifically, we introduce deep multi-modal structural equations, a generative model for causal effect estimation in which confounders are latent variables and unstructured data are proxy variables. This model supports multiple multi-modal proxies (images, text) as well as missing data. We empirically demonstrate that our approach outperforms existing methods based on propensity scores and corrects for confounding using unstructured inputs on tasks in genomics and healthcare. Our methods can potentially support the use of large amounts of data that were previously not used in causal inference.},
	language = {en},
	author = {Deshpande, Shachi and Wang, Kaiwen and Sreenivas, Dhruv and Li, Zheng and Kuleshov, Volodymyr},
	pages = {32},
	file = {Deshpande et al. - Deep Multi-Modal Structural Equations For Causal E.pdf:/Users/jacquelinemaasch/Zotero/storage/HERQ6L4I/Deshpande et al. - Deep Multi-Modal Structural Equations For Causal E.pdf:application/pdf},
}

@article{mey_causal_nodate,
	title = {Causal {Discovery} in {Time} {Series} {Data} {Using} {Causally} {Invariant} {Locally} {Linear} {Models}},
	abstract = {Identifying causal relationship is an often desired, but difficult, task, and generally only possible under specific assumptions. In this paper we are considering the task of identifying causal relationships between entities that have a temporal axis, as for example continuous measurements of different components within a complex machine. We introduce a locally linear model class that allows us to recover causal relationships, assuming that the process is locally linear, that we have access to observations in diverse environments and that the causal structure is invariant across the different environments. We validate the model in an idealized theoretical and two experimental settings.},
	language = {en},
	author = {Mey, Alexander},
	pages = {12},
	file = {Mey - Causal Discovery in Time Series Data Using Causall.pdf:/Users/jacquelinemaasch/Zotero/storage/ECEIHBGM/Mey - Causal Discovery in Time Series Data Using Causall.pdf:application/pdf},
}

@article{zhou_multiscale_nodate,
	title = {Multiscale {Pixel} {Spatiotemporal} {Information} {Flows}},
	abstract = {We develop a formal algorithmic framework to compute multiscale pixel spatiotemporal information flows which capture, in an unbiased manner, salient causal relationships between pixels across space and time. Real spatiotemporal dynamical systems such as cellular morphodynamics are inherently complex, nonlinear and evolve over time in response to feedbacks. This makes it highly challenging to directly model, simulate, or fit observed phenomena from first principle physics. Oftentimes neither the salient variables nor the key relationships are known a priori to include in a mathematical model. Even if a model was possible, we may be limited in our ability to sample the necessary information for exact system identification and verification. Alternatively, causal measures have been developed to identify potential causal relationships statistically from only observational timeseries. However such measures have largely only been studied for unstructured 1D timeseries where objects-of-interest have been pre-segmented and tracked over time. This restricts their application either to analyse general video dynamics, where individual objects are impossible to define or difficult to segment, or to understand potential causal relationships between subparts of objects. Here we propose a formal definition of a pixel spatiotemporal information flow as a spatiotemporal derivative of a pixel intensity timeseries to extract the dense pixel-to-pixel information transfer in 2D + time videos using any desired 1D causal measure, in a general and multiscale manner. Applying our framework, we discover salient pixel-to-pixel information highways in videos of diverse phenomena spanning traffic and crowd flow, collision physics, fish swarming, moving camouflaged animals, human action, embryo development, cell division and cell migration.},
	language = {en},
	author = {Zhou, Felix Y and Ravishankar, Roshan},
	pages = {14},
	file = {Zhou and Ravishankar - Multiscale Pixel Spatiotemporal Information Flows.pdf:/Users/jacquelinemaasch/Zotero/storage/5LB9JXF8/Zhou and Ravishankar - Multiscale Pixel Spatiotemporal Information Flows.pdf:application/pdf},
}

@misc{nguyen_glacial_2022,
	title = {{GLACIAL}: {Granger} and {Learning}-based {Causality} {Analysis} for {Longitudinal} {Studies}},
	shorttitle = {{GLACIAL}},
	url = {http://arxiv.org/abs/2210.07416},
	abstract = {The Granger framework is widely used for discovering causal relationships based on time-varying signals. Implementations of Granger causality (GC) are mostly developed for densely sampled timeseries data. A substantially different setting, particularly common in population health applications, is the longitudinal study design, where multiple individuals are followed and sparsely observed for a limited number of times. Longitudinal studies commonly track many variables, which are likely governed by nonlinear dynamics that might have individual-specific idiosyncrasies and exhibit both direct and indirect causes. Furthermore, real-world longitudinal data often suffer from widespread missingness. GC methods are not well-suited to handle these issues. In this paper, we intend to fill this methodological gap. We propose to marry the GC framework with a machine learning based prediction model. We call our approach GLACIAL, which stands for "Granger and LeArning-based CausalIty Analysis for Longitudinal studies." GLACIAL treats individuals as independent samples and uses average prediction accuracy on hold-out individuals to test for effects of causal relationships. GLACIAL employs a multi-task neural network trained with input feature dropout to efficiently learn nonlinear dynamic relationships between a large number of variables, handle missing values, and probe causal links. Extensive experiments on synthetic and real data demonstrate the utility of GLACIAL and how it can outperform competitive baselines.},
	language = {en},
	urldate = {2022-12-06},
	publisher = {arXiv},
	author = {Nguyen, Minh and Ngo, Gia H. and Sabuncu, Mert R.},
	month = oct,
	year = {2022},
	note = {arXiv:2210.07416 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Nguyen et al. - 2022 - GLACIAL Granger and Learning-based Causality Anal.pdf:/Users/jacquelinemaasch/Zotero/storage/4DWKUA27/Nguyen et al. - 2022 - GLACIAL Granger and Learning-based Causality Anal.pdf:application/pdf},
}

@incollection{hutchison_causal_2006,
	address = {Berlin, Heidelberg},
	title = {Causal {Discovery} with {Prior} {Information}},
	volume = {4304},
	isbn = {978-3-540-49787-5 978-3-540-49788-2},
	url = {http://link.springer.com/10.1007/11941439_141},
	abstract = {Bayesian networks (BNs) are rapidly becoming a leading tool in applied Artiﬁcial Intelligence (AI). BNs may be built by eliciting expert knowledge or learned via causal discovery programs. A hybrid approach is to incorporate prior information elicited from experts into the causal discovery process. We present several ways of using expert information as prior probabilities in the CaMML causal discovery program.},
	language = {en},
	urldate = {2022-12-08},
	booktitle = {{AI} 2006: {Advances} in {Artificial} {Intelligence}},
	publisher = {Springer Berlin Heidelberg},
	author = {O’Donnell, R. T. and Nicholson, A. E. and Han, B. and Korb, K. B. and Alam, M. J. and Hope, L. R.},
	editor = {Hutchison, David and Kanade, Takeo and Kittler, Josef and Kleinberg, Jon M. and Mattern, Friedemann and Mitchell, John C. and Naor, Moni and Nierstrasz, Oscar and Pandu Rangan, C. and Steffen, Bernhard and Sudan, Madhu and Terzopoulos, Demetri and Tygar, Dough and Vardi, Moshe Y. and Weikum, Gerhard and Sattar, Abdul and Kang, Byeong-ho},
	year = {2006},
	doi = {10.1007/11941439_141},
	note = {Series Title: Lecture Notes in Computer Science},
	pages = {1162--1167},
	file = {O’Donnell et al. - 2006 - Causal Discovery with Prior Information.pdf:/Users/jacquelinemaasch/Zotero/storage/WB4BZS26/O’Donnell et al. - 2006 - Causal Discovery with Prior Information.pdf:application/pdf},
}

@article{julia_flores_incorporating_2011,
	title = {Incorporating expert knowledge when learning {Bayesian} network structure: {A} medical case study},
	volume = {53},
	issn = {09333657},
	shorttitle = {Incorporating expert knowledge when learning {Bayesian} network structure},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0933365711001084},
	doi = {10.1016/j.artmed.2011.08.004},
	abstract = {Objectives: Bayesian networks (BNs) are rapidly becoming a leading technology in applied Artiﬁcial Intelligence, with many applications in medicine. Both automated learning of BNs and expert elicitation have been used to build these networks, but the potentially more useful combination of these two methods remains underexplored. In this paper we examine a number of approaches to their combination when learning structure and present new techniques for assessing their results.
Methods and materials: Using public-domain medical data, we run an automated causal discovery system, CaMML, which allows the incorporation of multiple kinds of prior expert knowledge into its search, to test and compare unbiased discovery with discovery biased with different kinds of expert opinion. We use adjacency matrices enhanced with numerical and colour labels to assist with the interpretation of the results. We present an algorithm for generating a single BN from a set of learned BNs that incorporates user preferences regarding complexity vs completeness. These techniques are presented as part of the ﬁrst detailed workﬂow for hybrid structure learning within the broader knowledge engineering process.
Results: The detailed knowledge engineering workﬂow is shown to be useful for structuring a complex iterative BN development process. The adjacency matrices make it clear that for our medical case study using the IOWA dataset, the simplest kind of prior information (partially sorting variables into tiers) was more effective in aiding model discovery than either using no prior information or using more sophisticated and detailed expert priors. The method for generating a single BN captures relationships that would be overlooked by other approaches in the literature.
Conclusion: Hybrid causal learning of BNs is an important emerging technology. We present methods for incorporating it into the knowledge engineering process, including visualisation and analysis of the learned networks.},
	language = {en},
	number = {3},
	urldate = {2022-12-08},
	journal = {Artificial Intelligence in Medicine},
	author = {Julia Flores, M. and Nicholson, Ann E. and Brunskill, Andrew and Korb, Kevin B. and Mascaro, Steven},
	month = nov,
	year = {2011},
	pages = {181--204},
	file = {Julia Flores et al. - 2011 - Incorporating expert knowledge when learning Bayes.pdf:/Users/jacquelinemaasch/Zotero/storage/VC39VHNB/Julia Flores et al. - 2011 - Incorporating expert knowledge when learning Bayes.pdf:application/pdf},
}

@article{melkas_interactive_nodate,
	title = {Interactive {Causal} {Structure} {Discovery} in {Earth} {System} {Sciences}},
	abstract = {Causal structure discovery (CSD) models are making inroads into several domains, including Earth system sciences. Their widespread adaptation is however hampered by the fact that the resulting models often do not take into account the domain knowledge of the experts and that it is often necessary to modify the resulting models iteratively. We present a workﬂow that is required to take this knowledge into account and to apply CSD algorithms in Earth system sciences. At the same time, we describe open research questions that still need to be addressed. We present a way to interactively modify the outputs of the CSD algorithms and argue that the user interaction can be modelled as a greedy ﬁnding of the local maximum-a-posteriori solution of the likelihood function, which is composed of the likelihood of the causal model and the prior distribution representing the knowledge of the expert user. We use a real-world data set for examples constructed in collaboration with our co-authors, who are the domain area experts. We show that ﬁnding maximally usable causal models in the Earth system sciences or other similar domains is a diﬃcult task which contains many interesting open research questions. We argue that taking the domain knowledge into account has a substantial eﬀect on the ﬁnal causal models discovered.},
	language = {en},
	author = {Melkas, Laila and Savvides, Rafael and Chandramouli, Suyog H and Makela, Jarmo and Nieminen, Tuomo and Mammarella, Ivan and Puolamaki, Kai},
	pages = {23},
	file = {Melkas et al. - Interactive Causal Structure Discovery in Earth Sy.pdf:/Users/jacquelinemaasch/Zotero/storage/R2Q3T2FM/Melkas et al. - Interactive Causal Structure Discovery in Earth Sy.pdf:application/pdf},
}

@incollection{heckerman_bayesian_2006,
	series = {Studies in {Fuzziness} and {Soft} {Computing}},
	title = {A {Bayesian} {Approach} to {Causal} {Discovery}},
	volume = {194},
	url = {https://doi.org/10.1007/3-540-33486-6_1},
	abstract = {We examine the Bayesian approach to the discovery of causal DAG models and compare it to the constraint-based approach. Both approaches rely on the Causal Markov condition, but the two differ significantly in theory and practice. An important difference between the approaches is that the constraint-based approach uses categorical information about conditional-independence constraints in the domain, whereas the Bayesian approach weighs the degree to which such constraints hold. As a result, the Bayesian approach has three distinct advantages over its constraint-based counterpart. One, conclusions derived from the Bayesian approach are not susceptible to incorrect categorical decisions about independence facts that can occur with data sets of finite size. Two, using the Bayesian approach, finer distinctions among model structures—both quantitative and qualitative—can be made. Three, information from several models can be combined to make better inferences and to better account for modeling uncertainty. In addition to describing the general Bayesian approach to causal discovery, we review approximation methods for missing data and hidden variables, and illustrate differences between the Bayesian and constraint-based methods using artificial and real examples.},
	language = {en},
	booktitle = {Innovations in {Machine} {Learning}},
	publisher = {Springer-Verlag},
	author = {Heckerman, David and Meek, Christopher and Cooper, Gregory},
	year = {2006},
	pages = {1--28},
	file = {Heckerman et al. - 2006 - A Bayesian Approach to Causal Discovery.pdf:/Users/jacquelinemaasch/Zotero/storage/YY43RAYW/Heckerman et al. - 2006 - A Bayesian Approach to Causal Discovery.pdf:application/pdf},
}

@article{deng_deep_nodate,
	title = {Deep {Causal} {Learning}: {Representation}, {Discovery} and {Inference}},
	language = {en},
	author = {Deng, Zizhen and Zheng, Xiaolong and Tian, Hu and Zeng, Daniel Dajun},
	pages = {35},
	file = {Deng et al. - Deep Causal Learning Representation, Discovery an.pdf:/Users/jacquelinemaasch/Zotero/storage/NMII9PB3/Deng et al. - Deep Causal Learning Representation, Discovery an.pdf:application/pdf},
}

@article{peters_causal_2016,
	title = {Causal inference by using invariant prediction: identification and confidence intervals},
	volume = {78},
	issn = {13697412},
	shorttitle = {Causal inference by using invariant prediction},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/rssb.12167},
	doi = {10.1111/rssb.12167},
	abstract = {What is the difference between a prediction that is made with a causal model and that with a non-causal model? Suppose that we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (e.g. various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid conﬁdence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufﬁcient assumptions under which the set of causal predictors becomes identiﬁable. We further investigate robustness properties of our approach under model misspeciﬁcation and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.},
	language = {en},
	number = {5},
	urldate = {2022-12-08},
	journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
	author = {Peters, Jonas and Bühlmann, Peter and Meinshausen, Nicolai},
	month = nov,
	year = {2016},
	pages = {947--1012},
	file = {Peters et al. - 2016 - Causal inference by using invariant prediction id.pdf:/Users/jacquelinemaasch/Zotero/storage/IDKK5XPD/Peters et al. - 2016 - Causal inference by using invariant prediction id.pdf:application/pdf},
}

@article{ghassami_learning_nodate,
	title = {Learning {Causal} {Structures} {Using} {Regression} {Invariance}},
	abstract = {We study causal discovery in a multi-environment setting, in which the functional relations for producing the variables from their direct causes remain the same across environments, while the distribution of exogenous noises may vary. We introduce the idea of using the invariance of the functional relations of the variables to their causes across a set of environments for structure learning. We deﬁne a notion of completeness for a causal inference algorithm in this setting and prove the existence of such algorithm by proposing the baseline algorithm. Additionally, we present an alternate algorithm that has signiﬁcantly improved computational and sample complexity compared to the baseline algorithm. Experiment results show that the proposed algorithm outperforms the other existing algorithms.},
	language = {en},
	author = {Ghassami, AmirEmad and Salehkaleybar, Saber and Kiyavash, Negar and Zhang, Kun},
	pages = {11},
	file = {Ghassami et al. - Learning Causal Structures Using Regression Invari.pdf:/Users/jacquelinemaasch/Zotero/storage/CALVTW6Q/Ghassami et al. - Learning Causal Structures Using Regression Invari.pdf:application/pdf},
}

@article{zhang_causal_2017,
	title = {Causal {Discovery} {Using} {Regression}-{Based} {Conditional} {Independence} {Tests}},
	volume = {31},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/10698},
	doi = {10.1609/aaai.v31i1.10698},
	abstract = {Conditional independence (CI) testing is an important tool in causal discovery. Generally, by using CI tests, a set of Markov equivalence classes w.r.t. the observed data can be estimated by checking whether each pair of variables x and y is d-separated, given a set of variables Z. Due to the curse of dimensionality, CI testing is often difﬁcult to return a reliable result for high-dimensional Z. In this paper, we propose a regression-based CI test to relax the test of x ⊥ y{\textbar}Z to simpler unconditional independence tests of x − f (Z) ⊥ y − g(Z), and x − f (Z) ⊥ Z or y − g(Z) ⊥ Z under the assumption that the data-generating procedure follows additive noise models (ANMs). When the ANM is identiﬁable, we prove that x − f (Z) ⊥ y − g(Z) ⇒ x ⊥ y{\textbar}Z. We also show that 1) f and g can be easily estimated by regression, 2) our test is more powerful than the state-of-the-art kernel CI tests, and 3) existing causal learning algorithms can infer much more causal directions by using the proposed method.},
	language = {en},
	number = {1},
	urldate = {2022-12-08},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Zhang, Hao and Zhou, Shuigeng and Zhang, Kun and Guan, Jihong},
	month = feb,
	year = {2017},
	file = {Zhang et al. - 2017 - Causal Discovery Using Regression-Based Conditiona.pdf:/Users/jacquelinemaasch/Zotero/storage/AHDQTGSR/Zhang et al. - 2017 - Causal Discovery Using Regression-Based Conditiona.pdf:application/pdf},
}

@article{runge_inferring_2019,
	title = {Inferring causation from time series in {Earth} system sciences},
	volume = {10},
	issn = {2041-1723},
	doi = {10.1038/s41467-019-10105-3},
	language = {en},
	number = {1},
	urldate = {2022-12-08},
	journal = {Nature Communications},
	author = {Runge, Jakob and Bathiany, Sebastian and Bollt, Erik and Camps-Valls, Gustau and Coumou, Dim and Deyle, Ethan and Glymour, Clark and Kretschmer, Marlene and Mahecha, Miguel D. and Muñoz-Marí, Jordi and van Nes, Egbert H. and Peters, Jonas and Quax, Rick and Reichstein, Markus and Scheffer, Marten and Schölkopf, Bernhard and Spirtes, Peter and Sugihara, George and Sun, Jie and Zhang, Kun and Zscheischler, Jakob},
	month = dec,
	year = {2019},
	pages = {2553},
	file = {Runge et al. - 2019 - Inferring causation from time series in Earth syst.pdf:/Users/jacquelinemaasch/Zotero/storage/D78YKF3P/Runge et al. - 2019 - Inferring causation from time series in Earth syst.pdf:application/pdf},
}

@article{wang_local_2022,
	title = {Local {Causal} {Discovery} in {Multiple} {Manipulated} {Datasets}},
	issn = {2162-237X, 2162-2388},
	url = {https://ieeexplore.ieee.org/document/9684022/},
	doi = {10.1109/TNNLS.2021.3139389},
	abstract = {We consider the problem of distinguishing direct causes from direct effects of a target variable of interest from multiple manipulated datasets with unknown manipulated variables and nonidentical data distributions. Recent studies have shown that datasets attained from manipulated experiments (i.e., manipulated data) contain richer causal information than observational data for causal structure learning. Thus, in this article, we propose a new algorithm, which makes full use of the interventional properties of a causal model to discover the direct causes and direct effects of a target variable from multiple datasets with different manipulations. It is more suited to real-world cases and is also a challenge to be addressed in this article. First, we apply the backward framework to learn parents and children (PC) of a given target from multiple manipulated datasets. Second, we orient some edges connected to the target in advance through the assumption that the target variable is not manipulated and then orient the remaining undirected edges by ﬁnding invariant V-structures from multiple datasets. Third, we analyze the correctness of the proposed algorithm. To the best of our knowledge, the proposed algorithm is the ﬁrst that can identify the local causal structure of a given target from multiple manipulated datasets with unknown manipulated variables. Experimental results on standard Bayesian networks validate the effectiveness of our algorithm.},
	language = {en},
	urldate = {2022-12-08},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	author = {Wang, Yunxia and Cao, Fuyuan and Yu, Kui and Liang, Jiye},
	year = {2022},
	pages = {1--13},
	file = {Wang et al. - 2022 - Local Causal Discovery in Multiple Manipulated Dat.pdf:/Users/jacquelinemaasch/Zotero/storage/UNFA5NH5/Wang et al. - 2022 - Local Causal Discovery in Multiple Manipulated Dat.pdf:application/pdf},
}

@article{wang_efficient_2022,
	title = {Efficient {Causal} {Structure} {Learning} from {Multiple} {Interventional} {Datasets} with {Unknown} {Targets}},
	volume = {36},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/20836},
	doi = {10.1609/aaai.v36i8.20836},
	abstract = {We consider the problem of reducing the false discovery rate in multiple high-dimensional interventional datasets under unknown targets. Traditional algorithms merged directly multiple causal graphs learned, which ignores the contradictions of different datasets, leading to lots of inconsistent directions of edges. For reducing the contradictory information, we propose a new algorithm, which ﬁrst learns an interventional Markov equivalence class (I-MEC) before merging multiple graphs. It utilizes the full power of the constraints available in interventional data and combines ideas from local learning, intervention, and search-and-score techniques in a principled and effective way in different intervention experiments. Speciﬁcally, local learning on multiple datasets is used to build a causal skeleton. Perfect intervention destroys some possible triangles, leading to the identiﬁcation of more possible V-structures. And then a theoretically correct I-MEC is learned. Search and scoring techniques based on the learned I-MEC further identify the remaining unoriented edges. Both theoretical analysis and experiments on benchmark Bayesian networks with the number of variables from 20 to 724 validate that the effectiveness of our algorithm in reducing the false discovery rate in high-dimensional interventional data.},
	language = {en},
	number = {8},
	urldate = {2022-12-08},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Wang, Yunxia and Cao, Fuyuan and Yu, Kui and Liang, Jiye},
	month = jun,
	year = {2022},
	pages = {8584--8593},
	file = {Wang et al. - 2022 - Efficient Causal Structure Learning from Multiple .pdf:/Users/jacquelinemaasch/Zotero/storage/D2KIG94H/Wang et al. - 2022 - Efficient Causal Structure Learning from Multiple .pdf:application/pdf},
}

@article{cooper_simple_1997,
	title = {A {Simple} {Constraint}-{Based} {Algorithm} for {Efﬁciently} {Mining} {Observational} {Databases} for {Causal} {Relationships}},
	abstract = {This paper presents a simple, efﬁcient computer-based method for discovering causal relationships from databases that contain observational data. Observational data is passively observed, as contrasted with experimental data. Most of the databases available for data mining are observational. There is great potential for mining such databases to discover causal relationships. We illustrate how observational data can constrain the causal relationships among measured variables, sometimes to the point that we can conclude that one variable is causing another variable. The presentation here is based on a constraint-based approach to causal discovery. A primary purpose of this paper is to present the constraint-based causal discovery method in the simplest possible fashion in order to (1) readily convey the basic ideas that underlie more complex constraint-based causal discovery techniques, and (2) permit interested readers to rapidly program and apply the method to their own databases, as a start toward using more elaborate causal discovery algorithms.},
	language = {en},
	journal = {Data Mining and Knowledge Discovery},
	author = {Cooper, Gregory F},
	year = {1997},
	pages = {22},
	file = {Cooper - 1997 - A Simple Constraint-Based Algorithm for Efﬁciently.pdf:/Users/jacquelinemaasch/Zotero/storage/37DU8CMY/Cooper - 1997 - A Simple Constraint-Based Algorithm for Efﬁciently.pdf:application/pdf},
}

@article{wang_prior-knowledge-driven_2020,
	title = {Prior-{Knowledge}-{Driven} {Local} {Causal} {Structure} {Learning} and {Its} {Application} on {Causal} {Discovery} {Between} {Type} 2 {Diabetes} and {Bone} {Mineral} {Density}},
	volume = {8},
	issn = {2169-3536},
	url = {https://ieeexplore.ieee.org/document/9095242/},
	doi = {10.1109/ACCESS.2020.2994936},
	abstract = {Type 2 diabetes (T2DM), one of the most prevalent chronic diseases, affects the glucose metabolism of the human body, which decreases the quantity of life and brings a heavy burden on social medical care. Patients with T2DM are more likely to suffer bone fragility fracture as diabetes affects bone mineral density (BMD). However, the discovery of the determinant factors of BMD in a medical way is expensive and time-consuming. In this paper, we propose a novel algorithm, Prior-Knowledge-driven local Causal structure Learning (PKCL), to discover the underlying causal mechanism between BMD and its factors from the clinical data. Since there exist limited data but redundant prior knowledge for medicine, PKCL adequately utilize the prior knowledge to mine the local causal structure for the target relationship. Combining the medical prior knowledge with the discovered causal relationships, PKCL can achieve more reliable results without long-standing medical statistical experiments. Extensive experiments are conducted on a newly provided clinical data set. The experimental study of PKCL on the data is proved to highly corresponding with existing medical knowledge, which demonstrates the superiority and effectiveness of PKCL. To illustrate the importance of prior knowledge, the result of the algorithm without prior knowledge is also investigated.},
	language = {en},
	urldate = {2022-12-08},
	journal = {IEEE Access},
	author = {Wang, Wei and Hu, Gangqiang and Yuan, Bo and Ye, Shandong and Chen, Chao and Cui, Yayun and Zhang, Xi and Qian, Liting},
	year = {2020},
	pages = {108798--108810},
	file = {Wang et al. - 2020 - Prior-Knowledge-Driven Local Causal Structure Lear.pdf:/Users/jacquelinemaasch/Zotero/storage/PDCCTFS9/Wang et al. - 2020 - Prior-Knowledge-Driven Local Causal Structure Lear.pdf:application/pdf},
}

@article{fragoso_bayesian_2018,
	title = {Bayesian {Model} {Averaging}: {A} {Systematic} {Review} and {Conceptual} {Classification}: {BMA}: {A} {Systematic} {Review}},
	volume = {86},
	issn = {03067734},
	shorttitle = {Bayesian {Model} {Averaging}},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/insr.12243},
	doi = {10.1111/insr.12243},
	abstract = {Bayesian model averaging (BMA) provides a coherent and systematic mechanism for accounting for model uncertainty. It can be regarded as an direct application of Bayesian inference to the problem of model selection, combined estimation and prediction. BMA produces a straightforward model choice criterion and less risky predictions. However, the application of BMA is not always straightforward, leading to diverse assumptions and situational choices on its different aspects. Despite the widespread application of BMA in the literature, there were not many accounts of these differences and trends besides a few landmark revisions in the late 1990s and early 2000s, therefore not accounting for advancements made in the last decades. In this work, we present an account of these developments through a careful content analysis of 820 articles in BMA published between 1996 and 2016. We also develop a conceptual classiﬁcation scheme to better describe this vast literature, understand its trends and future directions and provide guidance for the researcher interested in both the application and development of the methodology. The results of the classiﬁcation scheme and content review are then used to discuss the present and future of the BMA literature.},
	language = {en},
	number = {1},
	urldate = {2022-12-14},
	journal = {International Statistical Review},
	author = {Fragoso, Tiago M. and Bertoli, Wesley and Louzada, Francisco},
	month = apr,
	year = {2018},
	pages = {1--28},
	file = {Fragoso et al. - 2018 - Bayesian Model Averaging A Systematic Review and .pdf:/Users/jacquelinemaasch/Zotero/storage/U8QDEKJT/Fragoso et al. - 2018 - Bayesian Model Averaging A Systematic Review and .pdf:application/pdf},
}

@article{hinne_conceptual_2020,
	title = {A {Conceptual} {Introduction} to {Bayesian} {Model} {Averaging}},
	volume = {3},
	abstract = {Many statistical scenarios initially involve several candidate models that describe the data-generating process. Analysis often proceeds by first selecting the best model according to some criterion and then learning about the parameters of this selected model. Crucially, however, in this approach the parameter estimates are conditioned on the selected model, and any uncertainty about the model-selection process is ignored. An alternative is to learn the parameters for all candidate models and then combine the estimates according to the posterior probabilities of the associated models. This approach is known as Bayesian model averaging (BMA). BMA has several important advantages over all-or-none selection methods, but has been used only sparingly in the social sciences. In this conceptual introduction, we explain the principles of BMA, describe its advantages over all-or-none model selection, and showcase its utility in three examples: analysis of covariance, meta-analysis, and network analysis.},
	number = {2},
	journal = {Advances in Methods and Practices in Psychological Science},
	author = {Hinne, Max and Gronau, Quentin F},
	year = {2020},
	pages = {200--215},
	file = {Hinne and Gronau - A Conceptual Introduction to Bayesian Model Averag.pdf:/Users/jacquelinemaasch/Zotero/storage/AHTFEUPK/Hinne and Gronau - A Conceptual Introduction to Bayesian Model Averag.pdf:application/pdf},
}

@misc{castelletti_bayesian_2020,
	title = {Bayesian causal inference in probit graphical models},
	url = {http://arxiv.org/abs/2009.04795},
	abstract = {We consider a binary response which is potentially aﬀected by a set of continuous variables. Of special interest is the causal eﬀect on the response due to an intervention on a speciﬁc variable. The latter can be meaningfully determined on the basis of observational data through suitable assumptions on the data generating mechanism. In particular we assume that the joint distribution obeys the conditional independencies (Markov properties) inherent in a Directed Acyclic Graph (DAG), and the DAG is given a causal interpretation through the notion of interventional distribution. We propose a DAG-probit model where the response is generated by discretization through a random threshold of a continuous latent variable and the latter, jointly with the remaining continuous variables, has a distribution belonging to a zero-mean Gaussian model whose covariance matrix is constrained to satisfy the Markov properties of the DAG. Our model leads to a natural deﬁnition of causal eﬀect conditionally on a given DAG. Since the DAG which generates the observations is unknown, we present an eﬃcient MCMC algorithm whose target is the posterior distribution on the space of DAGs, the Cholesky parameters of the concentration matrix, and the threshold linking the response to the latent. Our end result is a Bayesian Model Averaging estimate of the causal eﬀect which incorporates parameter, as well as model, uncertainty. The methodology is assessed using simulation experiments and applied to a gene expression data set originating from breast cancer stem cells.},
	language = {en},
	urldate = {2022-12-14},
	publisher = {arXiv},
	author = {Castelletti, Federico and Consonni, Guido},
	month = sep,
	year = {2020},
	note = {arXiv:2009.04795 [stat]},
	keywords = {Statistics - Methodology},
	file = {Castelletti and Consonni - 2020 - Bayesian causal inference in probit graphical mode.pdf:/Users/jacquelinemaasch/Zotero/storage/FSDPFHVG/Castelletti and Consonni - 2020 - Bayesian causal inference in probit graphical mode.pdf:application/pdf},
}

@article{mani_theoretical_nodate,
	title = {A theoretical study of {Y} structures for causal discovery},
	abstract = {Causal discovery from observational data in the presence of unobserved variables is challenging. Identiﬁcation of so-called Y substructures is a suﬃcient condition for ascertaining some causal relations in the large sample limit, without the assumption of no hidden common causes. An example of a Y substructure is A → C, B → C, C → D. This paper describes the ﬁrst asymptotically reliable and computationally feasible scorebased search for discrete Y structures that does not assume that there are no unobserved common causes. For any parameterization of a directed acyclic graph (DAG) that has scores with the property that any DAG that can represent the distribution beats any DAG that can’t, and for two DAGs that represent the distribution, if one has fewer parameters than the other, the one with the fewest parameter wins. In this framework there is no need to assign scores to causal structures with unobserved common causes. The paper also describes how the existence of a Y structure shows the presence of an unconfounded causal relation, without assuming that there are no hidden common causes.},
	language = {en},
	author = {Mani, Subramani and Spirtes, Peter and Cooper, Gregory F},
	file = {Mani et al. - A theoretical study of Y structures for causal dis.pdf:/Users/jacquelinemaasch/Zotero/storage/HRCXUHUN/Mani et al. - A theoretical study of Y structures for causal dis.pdf:application/pdf},
}

@article{tian_bayesian_nodate,
	title = {Bayesian {Model} {Averaging} {Using} the k-best {Bayesian} {Network} {Structures}},
	abstract = {We study the problem of learning Bayesian network structures from data. We develop an algorithm for ﬁnding the k-best Bayesian network structures. We propose to compute the posterior probabilities of hypotheses of interest by Bayesian model averaging over the k-best Bayesian networks. We present empirical results on structural discovery over several real and synthetic data sets and show that the method outperforms the model selection method and the stateof-the-art MCMC methods.},
	language = {en},
	author = {Tian, Jin and He, Ru and Ram, Lavanya},
	file = {Tian et al. - Bayesian Model Averaging Using the k-best Bayesian.pdf:/Users/jacquelinemaasch/Zotero/storage/KCZ92R7N/Tian et al. - Bayesian Model Averaging Using the k-best Bayesian.pdf:application/pdf},
}

@incollection{tadesse_bayesian_2021,
	address = {Boca Raton},
	edition = {1},
	title = {Bayesian {Model} {Averaging} in {Causal} {Inference}},
	isbn = {978-1-00-308901-8},
	url = {https://www.taylorfrancis.com/books/9781003089018/chapters/10.1201/9781003089018-9},
	language = {en},
	urldate = {2022-12-14},
	booktitle = {Handbook of {Bayesian} {Variable} {Selection}},
	publisher = {Chapman and Hall/CRC},
	author = {Antonelli, Joseph and Dominici, Francesca},
	collaborator = {Tadesse, Mahlet G. and Vannucci, Marina},
	month = dec,
	year = {2021},
	doi = {10.1201/9781003089018-9},
	pages = {201--226},
	file = {Antonelli and Dominici - 2021 - Bayesian Model Averaging in Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/49J2NF4B/Antonelli and Dominici - 2021 - Bayesian Model Averaging in Causal Inference.pdf:application/pdf},
}

@article{talbot_generalized_2022,
	title = {A generalized double robust {Bayesian} model averaging approach to causal effect estimation with application to the study of osteoporotic fractures},
	volume = {10},
	issn = {2193-3685},
	url = {https://www.degruyter.com/document/doi/10.1515/jci-2021-0023/html},
	doi = {10.1515/jci-2021-0023},
	abstract = {Analysts often use data-driven approaches to supplement their knowledge when selecting covariates for eﬀect estimation. Multiple variable selection procedures for causal eﬀect estimation have been devised in recent years, but additional developments are still required to adequately address the needs of analysts. We propose a generalized Bayesian causal eﬀect estimation (GBCEE) algorithm to perform variable selection and produce double robust (DR) estimates of causal eﬀects for binary or continuous exposures and outcomes. GBCEE employs a prior distribution that targets the selection of true confounders and predictors of the outcome for the unbiased estimation of causal eﬀects with reduced standard errors. The Bayesian machinery allows GBCEE to directly produce inferences for its estimate. In simulations, GBCEE was observed to perform similarly or to outperform DR alternatives. Its ability to directly produce inferences is also an important advantage from a computational perspective. The method is ﬁnally illustrated for the estimation of the eﬀect of meeting physical activity recommendations on the risk of hip or upper-leg fractures among older women in the study of osteoporotic fractures. The 95\% conﬁdence interval produced by GBCEE is 61\% narrower than that of a DR estimator adjusting for all potential confounders in this illustration.},
	language = {en},
	number = {1},
	urldate = {2022-12-14},
	journal = {Journal of Causal Inference},
	author = {Talbot, Denis and Beaudoin, Claudia},
	month = nov,
	year = {2022},
	pages = {335--371},
	file = {Talbot and Beaudoin - 2022 - A generalized double robust Bayesian model averagi.pdf:/Users/jacquelinemaasch/Zotero/storage/ZS8YTUIU/Talbot and Beaudoin - 2022 - A generalized double robust Bayesian model averagi.pdf:application/pdf},
}

@article{hinne_bayesian_2022,
	title = {Bayesian model averaging for nonparametric discontinuity design},
	volume = {17},
	issn = {1932-6203},
	url = {https://dx.plos.org/10.1371/journal.pone.0270310},
	doi = {10.1371/journal.pone.0270310},
	abstract = {Quasi-experimental research designs, such as regression discontinuity and interrupted time series, allow for causal inference in the absence of a randomized controlled trial, at the cost of additional assumptions. In this paper, we provide a framework for discontinuity-based designs using Bayesian model averaging and Gaussian process regression, which we refer to as ‘Bayesian nonparametric discontinuity design’, or BNDD for short. BNDD addresses the two major shortcomings in most implementations of such designs: overconfidence due to implicit conditioning on the alleged effect, and model misspecification due to reliance on overly simplistic regression models. With the appropriate Gaussian process covariance function, our approach can detect discontinuities of any order, and in spectral features. We demonstrate the usage of BNDD in simulations, and apply the framework to determine the effect of running for political positions on longevity, of the effect of an alleged historical phantom border in the Netherlands on Dutch voting behaviour, and of Kundalini Yoga meditation on heart rate.},
	language = {en},
	number = {6},
	urldate = {2022-12-14},
	journal = {PLOS ONE},
	author = {Hinne, Max and Leeftink, David and van Gerven, Marcel A. J. and Ambrogioni, Luca},
	editor = {Saikia, Lalit Chandra},
	month = jun,
	year = {2022},
	pages = {e0270310},
	file = {Hinne et al. - 2022 - Bayesian model averaging for nonparametric discont.pdf:/Users/jacquelinemaasch/Zotero/storage/Z7WBK28U/Hinne et al. - 2022 - Bayesian model averaging for nonparametric discont.pdf:application/pdf},
}

@misc{kleinegesse_domain_2022,
	title = {Domain {Knowledge} in {A}*-{Based} {Causal} {Discovery}},
	url = {http://arxiv.org/abs/2208.08247},
	abstract = {Causal discovery has become a vital tool for scientists and practitioners wanting to discover causal relationships from observational data. While most previous approaches to causal discovery have implicitly assumed that no expert domain knowledge is available, practitioners can often provide such domain knowledge from prior experience. Recent work has incorporated domain knowledge into constraint-based causal discovery. The majority of such constraint-based methods, however, assume causal faithfulness, which has been shown to be frequently violated in practice. Consequently, there has been renewed attention towards exact-search score-based causal discovery methods, which do not assume causal faithfulness, such as A*-based methods. However, there has been no consideration of these methods in the context of domain knowledge. In this work, we focus on efﬁciently integrating several types of domain knowledge into A*-based causal discovery. In doing so, we discuss and explain how domain knowledge can reduce the graph search space and then provide an analysis of the potential computational gains. We support these ﬁndings with experiments on synthetic and real data, showing that even small amounts of domain knowledge can dramatically speed up A*-based causal discovery and improve its performance and practicality.},
	language = {en},
	urldate = {2022-12-14},
	publisher = {arXiv},
	author = {Kleinegesse, Steven and Lawrence, Andrew R. and Chockler, Hana},
	month = aug,
	year = {2022},
	note = {arXiv:2208.08247 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence, 68T99},
	file = {Kleinegesse et al. - 2022 - Domain Knowledge in A-Based Causal Discovery.pdf:/Users/jacquelinemaasch/Zotero/storage/6NL7HN3H/Kleinegesse et al. - 2022 - Domain Knowledge in A-Based Causal Discovery.pdf:application/pdf},
}

@misc{dong_high-dimensional_2022,
	title = {High-{Dimensional} {Causal} {Discovery}: {Learning} from {Inverse} {Covariance} via {Independence}-based {Decomposition}},
	shorttitle = {High-{Dimensional} {Causal} {Discovery}},
	url = {http://arxiv.org/abs/2211.14221},
	abstract = {Inferring causal relationships from observational data is a fundamental yet highly complex problem when the number of variables is large. Recent advances have made much progress in learning causal structure models (SEMs) but still face challenges in scalability. This paper aims to eﬃciently discover causal DAGs from high-dimensional data. We investigate a way of recovering causal DAGs from inverse covariance estimators of the observational data. The proposed algorithm, called ICID (inverse covariance estimation and independence-based decomposition), searches for a decomposition of the inverse covariance matrix that preserves its nonzero patterns. This algorithm beneﬁts from properties of positive deﬁnite matrices supported on chordal graphs and the preservation of nonzero patterns in their Cholesky decomposition; we ﬁnd exact mirroring between the support-preserving property and the independence-preserving property of our decomposition method, which explains its eﬀectiveness in identifying causal structures from the data distribution. We show that the proposed algorithm recovers causal DAGs with a complexity of O(d2) in the context of sparse SEMs. The advantageously low complexity is reﬂected by good scalability of our algorithm in thorough experiments and comparisons with state-of-the-art algorithms.},
	language = {en},
	urldate = {2022-12-14},
	publisher = {arXiv},
	author = {Dong, Shuyu and Uemura, Kento and Fujii, Akito and Chang, Shuang and Koyanagi, Yusuke and Maruhashi, Koji and Sebag, Michèle},
	month = nov,
	year = {2022},
	note = {arXiv:2211.14221 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology, Mathematics - Optimization and Control},
	file = {Dong et al. - 2022 - High-Dimensional Causal Discovery Learning from I.pdf:/Users/jacquelinemaasch/Zotero/storage/F3NS6LWY/Dong et al. - 2022 - High-Dimensional Causal Discovery Learning from I.pdf:application/pdf},
}

@article{consonni_prior_2018,
	title = {Prior {Distributions} for {Objective} {Bayesian} {Analysis}},
	volume = {13},
	issn = {1936-0975},
	url = {https://projecteuclid.org/journals/bayesian-analysis/volume-13/issue-2/Prior-Distributions-for-Objective-Bayesian-Analysis/10.1214/18-BA1103.full},
	doi = {10.1214/18-BA1103},
	abstract = {We provide a review of prior distributions for objective Bayesian analysis. We start by examining some foundational issues and then organize our exposition into priors for: i) estimation or prediction; ii) model selection; iii) highdimensional models. With regard to i), we present some basic notions, and then move to more recent contributions on discrete parameter space, hierarchical models, nonparametric models, and penalizing complexity priors. Point ii) is the focus of this paper: it discusses principles for objective Bayesian model comparison, and singles out some major concepts for building priors, which are subsequently illustrated in some detail for the classic problem of variable selection in normal linear models. We also present some recent contributions in the area of objective priors on model space. With regard to point iii) we only provide a short summary of some default priors for high-dimensional models, a rapidly growing area of research.},
	language = {en},
	number = {2},
	urldate = {2022-12-14},
	journal = {Bayesian Analysis},
	author = {Consonni, Guido and Fouskakis, Dimitris and Liseo, Brunero and Ntzoufras, Ioannis},
	month = jun,
	year = {2018},
	file = {Consonni et al. - 2018 - Prior Distributions for Objective Bayesian Analysi.pdf:/Users/jacquelinemaasch/Zotero/storage/JLPBBYQU/Consonni et al. - 2018 - Prior Distributions for Objective Bayesian Analysi.pdf:application/pdf},
}

@article{kaplan_bayesian_2016,
	title = {Bayesian {Model} {Averaging} {Over} {Directed} {Acyclic} {Graphs} {With} {Implications} for the {Predictive} {Performance} of {Structural} {Equation} {Models}},
	volume = {23},
	issn = {1070-5511, 1532-8007},
	url = {http://www.tandfonline.com/doi/full/10.1080/10705511.2015.1092088},
	doi = {10.1080/10705511.2015.1092088},
	abstract = {This article examines Bayesian model averaging as a means of addressing predictive performance in Bayesian structural equation models. The current approach to addressing the problem of model uncertainty lies in the method of Bayesian model averaging. We expand the work of Madigan and his colleagues by considering a structural equation model as a special case of a directed acyclic graph. We then provide an algorithm that searches the model space for submodels and obtains a weighted average of the submodels using posterior model probabilities as weights. Our simulation study provides a frequentist evaluation of our Bayesian model averaging approach and indicates that when the true model is known, Bayesian model averaging does not yield necessarily better predictive performance compared to nonaveraged models. However, our case study using data from an international large-scale assessment reveals that the model-averaged submodels provide better posterior predictive performance compared to the initially speciﬁed model.},
	language = {en},
	number = {3},
	urldate = {2022-12-14},
	journal = {Structural Equation Modeling: A Multidisciplinary Journal},
	author = {Kaplan, David and Lee, Chansoon},
	month = may,
	year = {2016},
	pages = {343--353},
	file = {Kaplan and Lee - 2016 - Bayesian Model Averaging Over Directed Acyclic Gra.pdf:/Users/jacquelinemaasch/Zotero/storage/MICETKMY/Kaplan and Lee - 2016 - Bayesian Model Averaging Over Directed Acyclic Gra.pdf:application/pdf},
}

@article{penny_comparing_2010,
	title = {Comparing {Families} of {Dynamic} {Causal} {Models}},
	volume = {6},
	issn = {1553-7358},
	url = {https://dx.plos.org/10.1371/journal.pcbi.1000709},
	doi = {10.1371/journal.pcbi.1000709},
	abstract = {Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This ‘‘best model’’ approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data.},
	language = {en},
	number = {3},
	urldate = {2022-12-14},
	journal = {PLoS Computational Biology},
	author = {Penny, Will D. and Stephan, Klaas E. and Daunizeau, Jean and Rosa, Maria J. and Friston, Karl J. and Schofield, Thomas M. and Leff, Alex P.},
	editor = {Kording, Konrad P.},
	month = mar,
	year = {2010},
	pages = {e1000709},
	file = {Penny et al. - 2010 - Comparing Families of Dynamic Causal Models.pdf:/Users/jacquelinemaasch/Zotero/storage/NBYZK23F/Penny et al. - 2010 - Comparing Families of Dynamic Causal Models.pdf:application/pdf},
}

@article{ogarrio_hybrid_nodate,
	title = {A {Hybrid} {Causal} {Search} {Algorithm} for {Latent} {Variable} {Models}},
	abstract = {Existing score-based causal model search algorithms such as GES (and a speeded up version, FGS) are asymptotically correct, fast, and reliable, but make the unrealistic assumption that the true causal graph does not contain any unmeasured confounders. There are several constraint-based causal search algorithms (e.g RFCI, FCI, or FCI+) that are asymptotically correct without assuming that there are no unmeasured confounders, but often perform poorly on small samples. We describe a combined score and constraint-based algorithm, GFCI, that we prove is asymptotically correct. On synthetic data, GFCI is only slightly slower than RFCI but more accurate than FCI, RFCI and FCI+.},
	language = {en},
	author = {Ogarrio, Juan Miguel and Spirtes, Peter and Ramsey, Joe},
	file = {Ogarrio et al. - A Hybrid Causal Search Algorithm for Latent Variab.pdf:/Users/jacquelinemaasch/Zotero/storage/S6WH7W3P/Ogarrio et al. - A Hybrid Causal Search Algorithm for Latent Variab.pdf:application/pdf},
}

@article{lei_causal_nodate,
	title = {Causal {Discovery} for {Modular} {World} {Models}},
	abstract = {Latent world models allow agents to reason about complex environments with high-dimensional observations. However, adapting to new environments and effectively leveraging previous knowledge remain significant challenges. We present variational causal dynamics (VCD), a structured world model that exploits the invariance of causal mechanisms across environments to achieve fast and modular adaptation. VCD identifies reusable components across different environments by combining causal discovery and variational inference to learn a latent representation and transition model jointly in an unsupervised manner. In evaluations on simulated environments with image observations, we show that VCD is able to successfully identify causal variables. Moreover, given a small number of observations in a previously unseen, intervened environment, VCD is able to identify the sparse changes in the dynamics and to adapt efficiently. In doing so, VCD significantly extends the capabilities of the current state-of-the-art in latent world models.},
	language = {en},
	author = {Lei, Anson and Schölkopf, Bernhard and Posner, Ingmar},
	file = {Lei et al. - Causal Discovery for Modular World Models.pdf:/Users/jacquelinemaasch/Zotero/storage/HH3JFGRS/Lei et al. - Causal Discovery for Modular World Models.pdf:application/pdf},
}

@article{li_deconfounding_2022,
	title = {Deconfounding {Physical} {Dynamics} with {Global} {Causal} {Relation} and {Confounder} {Transmission} for {Counterfactual} {Prediction}},
	volume = {36},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/20044},
	doi = {10.1609/aaai.v36i2.20044},
	abstract = {Discovering the underneath causal relations is the fundamental ability for reasoning about the surrounding environment and predicting the future states in the physical world. Counterfactual prediction from visual input, which requires simulating future states based on unrealized situations in the past, is a vital component in causal relation tasks. In this paper, we work on the confounders that have effect on the physical dynamics, including masses, friction coefﬁcients, etc., to bridge relations between the intervened variable and the affected variable whose future state may be altered. We propose a neural network framework combining Global Causal Relation Attention (GCRA) and Confounder Transmission Structure (CTS). The GCRA looks for the latent causal relations between different variables and estimates the confounders by capturing both spatial and temporal information. The CTS integrates and transmits the learnt confounders in a residual way, so that the estimated confounders can be encoded into the network as a constraint for object positions when performing counterfactual prediction. Without any access to ground truth information about confounders, our model outperforms the state-of-the-art method on various benchmarks by fully utilizing the constraints of confounders. Extensive experiments demonstrate that our model can generalize to unseen environments and maintain good performance.},
	language = {en},
	number = {2},
	urldate = {2022-12-19},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Li, Zongzhao and Zhu, Xiangyu and Lei, Zhen and Zhang, Zhaoxiang},
	month = jun,
	year = {2022},
	pages = {1536--1545},
	file = {Li et al. - 2022 - Deconfounding Physical Dynamics with Global Causal.pdf:/Users/jacquelinemaasch/Zotero/storage/CNZUQA5B/Li et al. - 2022 - Deconfounding Physical Dynamics with Global Causal.pdf:application/pdf},
}

@inproceedings{bello2022dagma,
    author = {Bello, Kevin and Aragam, Bryon and Ravikumar, Pradeep},
    booktitle = {Advances in Neural Information Processing Systems},
    title = {{DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization}},
    year = {2022}
}

@article{chickering_learning_nodate,
	title = {Learning {Equivalence} {Classes} of {Bayesian} {Network} {Structures}},
	abstract = {Approaches to learning Bayesian networks from data typically combine a scoring func­ tion with a heuristic search procedure. Given a Bayesian network structure, many of the scoring functions derived in the literature re­ turn a score for the entire equivalence class to which the structure belongs. When using such a scoring function, it is appropriate for the heuristic search algorithm to search over equivalence classes of Bayesian networks as opposed to individual structures. We present the general formulation of a search space for which the states of the search correspond to equivalence classes of structures. Using this space, any one of a number of heuristic search algorithms can easily be applied. We com­ pare greedy search performance in the pro­ posed search space to greedy search perfor­ mance in a search space for which the states correspond to individual Bayesian network structures.},
	language = {en},
    journal = {Journal of Machine Learning Research},
	author = {Chickering, David Maxwell},
	file = {Chickering - Learning Equivalence Classes of Bayesian Network S.pdf:/Users/jacquelinemaasch/Zotero/storage/QZMEFLZJ/Chickering - Learning Equivalence Classes of Bayesian Network S.pdf:application/pdf},
year = {2013}
}

@article{ParkKim2020,
  author    = {Gunwoong Park and Youngwhan Kim},
  title     = {Identifiability of Gaussian linear structural equation models with homogeneous and heterogeneous error variances},
  journal   = {Journal of the Korean Statistical Society},
  volume    = {49},
  pages     = {276--292},
  year      = {2020},
  publisher = {Springer},
  doi       = {10.1007/s42952-020-00018-7}
}


@article{colombo2012learning,
  title={Learning high-dimensional directed acyclic graphs with latent and selection variables},
  author={Colombo, Diego and Maathuis, Marloes H. and Kalisch, Markus and Richardson, Thomas S.},
  journal={The Annals of Statistics},
  volume={40},
  number={1},
  pages={294--321},
  year={2012},
  publisher={Institute of Mathematical Statistics}
}

@inproceedings{claassen2013learning,
  title={Learning sparse causal models is not NP-hard},
  author={Claassen, Tom and Mooij, Joris M. and Heskes, Tom},
  booktitle={Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI-2013)},
  pages={172--181},
  year={2013},
  organization={AUAI Press}
}


@article{chickering2004large,
  title={Large-Sample Learning of Bayesian Networks is NP-Hard},
  author={Chickering, David Maxwell and Meek, Christopher and Heckerman, David},
  journal={Journal of Machine Learning Research},
  volume={5},
  pages={1287--1330},
  year={2004},
  publisher={JMLR. org}
}

@inproceedings{ganian2024revisiting,
  title={Revisiting Causal Discovery from a Complexity-Theoretic Perspective},
  author={Ganian, Robert and Korchemna, Viktoriia and Szeider, Stefan},
  booktitle={Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence (IJCAI-24)},
  pages={3377--3385},
  year={2024},
  organization={International Joint Conferences on Artificial Intelligence Organization}
}

@inproceedings{watson_causal_2022,
	title = {Causal {Discovery} {Under} a {Confounder} {Blanket}},
	url = {https://arxiv-org.proxy.library.cornell.edu/abs/2205.05715},
	abstract = {Inferring causal relationships from observational data is rarely straightforward, but the problem is especially difﬁcult in high dimensions. For these applications, causal discovery algorithms typically require parametric restrictions or extreme sparsity constraints. We relax these assumptions and focus on an important but more specialized problem, namely recovering the causal order among a subgraph of variables known to descend from some (possibly large) set of confounding covariates, i.e. a confounder blanket. This is useful in many settings, for example when studying a dynamic biomolecular subsystem with genetic data providing background information. Under a structural assumption called the confounder blanket principle, which we argue is essential for tractable causal discovery in high dimensions, our method accommodates graphs of low or high sparsity while maintaining polynomial time complexity. We present a structure learning algorithm that is provably sound and complete with respect to a so-called lazy oracle. We design inference procedures with ﬁnite sample error control for linear and nonlinear systems, and demonstrate our approach on a range of simulated and real-world datasets. An accompanying R package, cbl, is available from CRAN.},
	booktitle = {Proceedings of the 38th {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Watson, David S and Silva, Ricardo},
	year = {2022},
	pages = {2096--2106},
	file = {Watson and Silva - 2022 - Causal Discovery Under a Confounder Blanket.pdf:/Users/jacquelinemaasch/Zotero/storage/EA78D2J4/Watson and Silva - 2022 - Causal Discovery Under a Confounder Blanket.pdf:application/pdf},
}

@article{vanderweele_principles_2019,
	title = {Principles of confounder selection},
	volume = {34},
	issn = {0393-2990, 1573-7284},
	url = {https://link.springer.com/10.1007/s10654-019-00494-6},
	doi = {10.1007/s10654-019-00494-6},
	abstract = {Selecting an appropriate set of confounders for which to control is critical for reliable causal inference. Recent theoretical and methodological developments have helped clarify a number of principles of confounder selection. When complete knowledge of a causal diagram relating all covariates to each other is available, graphical rules can be used to make decisions about covariate control. Unfortunately, such complete knowledge is often unavailable. This paper puts forward a practical approach to confounder selection decisions when the somewhat less stringent assumption is made that knowledge is available for each covariate whether it is a cause of the exposure, and whether it is a cause of the outcome. Based on recent theoretically justified developments in the causal inference literature, the following proposal is made for covariate control decisions: control for each covariate that is a cause of the exposure, or of the outcome, or of both; exclude from this set any variable known to be an instrumental variable; and include as a covariate any proxy for an unmeasured variable that is a common cause of both the exposure and the outcome. Various principles of confounder selection are then further related to statistical covariate selection methods.},
	language = {en},
	number = {3},
	urldate = {2022-12-31},
	journal = {European Journal of Epidemiology},
	author = {VanderWeele, Tyler J.},
	month = mar,
	year = {2019},
	pages = {211--219},
	file = {VanderWeele - 2019 - Principles of confounder selection.pdf:/Users/jacquelinemaasch/Zotero/storage/3TUF5FLD/VanderWeele - 2019 - Principles of confounder selection.pdf:application/pdf},
}

@article{williamson_introduction_2014,
	title = {Introduction to causal diagrams for confounder selection: {Causal} diagrams},
	volume = {19},
	issn = {13237799},
	shorttitle = {Introduction to causal diagrams for confounder selection},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/resp.12238},
	doi = {10.1111/resp.12238},
	language = {en},
	number = {3},
	urldate = {2022-12-31},
	journal = {Respirology},
	author = {Williamson, Elizabeth J. and Aitken, Zoe and Lawrie, Jock and Dharmage, Shyamali C. and Burgess, John A. and Forbes, Andrew B.},
	month = apr,
	year = {2014},
	pages = {303--311},
	file = {Williamson et al. - 2014 - Introduction to causal diagrams for confounder sel.pdf:/Users/jacquelinemaasch/Zotero/storage/9FT2Q93R/Williamson et al. - 2014 - Introduction to causal diagrams for confounder sel.pdf:application/pdf},
}

@article{haggstrom_data-driven_2018,
	title = {Data-driven confounder selection via {Markov} and {Bayesian} networks},
	volume = {74},
	issn = {0006341X},
	shorttitle = {Data-driven confounder selection via {Markov} and {Bayesian} networks},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12788},
	doi = {10.1111/biom.12788},
	abstract = {To unbiasedly estimate a causal eﬀect on an outcome unconfoundedness is often assumed. If there is suﬃcient knowledge on the underlying causal structure then existing confounder selection criteria can be used to select subsets of the observed pretreatment covariates, X, suﬃcient for unconfoundedness, if such subsets exist. Here, estimation of these target subsets is considered when the underlying causal structure is unknown. The proposed method is to model the causal structure by a probabilistic graphical model, for example, a Markov or Bayesian network, estimate this graph from observed data and select the target subsets given the estimated graph. The approach is evaluated by simulation both in a high-dimensional setting where unconfoundedness holds given X and in a setting where unconfoundedness only holds given subsets of X. Several common target subsets are investigated and the selected subsets are compared with respect to accuracy in estimating the average causal eﬀect. The proposed method is implemented with existing software that can easily handle high-dimensional data, in terms of large samples and large number of covariates. The results from the simulation study show that, if unconfoundedness holds given X, this approach is very successful in selecting the target subsets, outperforming alternative approaches based on random forests and LASSO, and that the subset estimating the target subset containing all causes of outcome yields smallest MSE in the average causal eﬀect estimation.},
	language = {en},
	number = {2},
	urldate = {2022-12-31},
	journal = {Biometrics},
	author = {Häggström, Jenny},
	month = jun,
	year = {2018},
	pages = {389--398},
	file = {Häggström - 2018 - Data-driven confounder selection via Markov and Ba.pdf:/Users/jacquelinemaasch/Zotero/storage/FXDVNW8Q/Häggström - 2018 - Data-driven confounder selection via Markov and Ba.pdf:application/pdf},
}

@misc{guo_confounder_2022,
	title = {Confounder {Selection}: {Objectives} and {Approaches}},
	shorttitle = {Confounder {Selection}},
	url = {http://arxiv.org/abs/2208.13871},
	abstract = {Confounder selection is perhaps the most important step in the design of observational studies. A number of criteria, often with diﬀerent objectives and approaches, have been proposed, and their validity and practical value have been debated in the literature. Here, we provide a uniﬁed review of these criteria and the assumptions behind them. We list several objectives that confounder selection methods aim to achieve and discuss the amount of structural knowledge required by diﬀerent approaches. Finally, we discuss limitations of the existing approaches and implications for practitioners.},
	language = {en},
	urldate = {2022-12-31},
	publisher = {arXiv},
	author = {Guo, F. Richard and Lundborg, Anton Rask and Zhao, Qingyuan},
	month = aug,
	year = {2022},
	note = {arXiv:2208.13871 [math, stat]},
	keywords = {Statistics - Methodology, Mathematics - Statistics Theory},
	file = {Guo et al. - 2022 - Confounder Selection Objectives and Approaches.pdf:/Users/jacquelinemaasch/Zotero/storage/XEZELM3T/Guo et al. - 2022 - Confounder Selection Objectives and Approaches.pdf:application/pdf},
}

@article{wilson_confounder_2014,
	title = {Confounder selection via penalized credible regions: {Confounder} {Selection} via {Penalized} {Credible} {Regions}},
	volume = {70},
	issn = {0006341X},
	shorttitle = {Confounder selection via penalized credible regions},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12203},
	doi = {10.1111/biom.12203},
	abstract = {When estimating the eﬀect of an exposure or treatment on an outcome it is important to select the proper subset of confounding variables to include in the model. Including too many covariates increases mean square error on the eﬀect of interest while not including confounding variables biases the exposure eﬀect estimate. We propose a decision-theoretic approach to confounder selection and eﬀect estimation. We ﬁrst estimate the full standard Bayesian regression model and then post-process the posterior distribution with a loss function that penalizes models omitting important confounders. Our method can be ﬁt easily with existing software and in many situations without the use of Markov chain Monte Carlo methods, resulting in computation on the order of the least squares solution. We prove that the proposed estimator has attractive asymptotic properties. In a simulation study we show that our method outperforms existing methods. We demonstrate our method by estimating the eﬀect of ﬁne particulate matter (PM2.5) exposure on birth weight in Mecklenburg County, North Carolina.},
	language = {en},
	number = {4},
	urldate = {2022-12-31},
	journal = {Biometrics},
	author = {Wilson, Ander and Reich, Brian J.},
	month = dec,
	year = {2014},
	pages = {852--861},
	file = {Wilson and Reich - 2014 - Confounder selection via penalized credible region.pdf:/Users/jacquelinemaasch/Zotero/storage/B8PSLBGX/Wilson and Reich - 2014 - Confounder selection via penalized credible region.pdf:application/pdf},
}

@article{tennant_use_2021,
	title = {Use of directed acyclic graphs ({DAGs}) to identify confounders in applied health research: review and recommendations},
	volume = {50},
	issn = {0300-5771, 1464-3685},
	shorttitle = {Use of directed acyclic graphs ({DAGs}) to identify confounders in applied health research},
	url = {https://academic.oup.com/ije/article/50/2/620/6012812},
	doi = {10.1093/ije/dyaa213},
	abstract = {Background: Directed acyclic graphs (DAGs) are an increasingly popular approach for identifying confounding variables that require conditioning when estimating causal effects. This review examined the use of DAGs in applied health research to inform recommendations for improving their transparency and utility in future research.},
	language = {en},
	number = {2},
	urldate = {2022-12-31},
	journal = {International Journal of Epidemiology},
	author = {Tennant, Peter W G and Murray, Eleanor J and Arnold, Kellyn F and Berrie, Laurie and Fox, Matthew P and Gadd, Sarah C and Harrison, Wendy J and Keeble, Claire and Ranker, Lynsie R and Textor, Johannes and Tomova, Georgia D and Gilthorpe, Mark S and Ellison, George T H},
	month = may,
	year = {2021},
	pages = {620--632},
	file = {Tennant et al. - 2021 - Use of directed acyclic graphs (DAGs) to identify .pdf:/Users/jacquelinemaasch/Zotero/storage/Y9CK9FQ9/Tennant et al. - 2021 - Use of directed acyclic graphs (DAGs) to identify .pdf:application/pdf},
}

@article{vanderweele_new_2011,
	title = {A {New} {Criterion} for {Confounder} {Selection}},
	volume = {67},
	issn = {0006341X},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/j.1541-0420.2011.01619.x},
	doi = {10.1111/j.1541-0420.2011.01619.x},
	abstract = {We propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available. We assume all covariates being considered are pretreatment variables and that for each covariate it is known (i) whether the covariate is a cause of treatment, and (ii) whether the covariate is a cause of the outcome. The causal relationships the covariates have with one another is assumed unknown. We propose that control be made for any covariate that is either a cause of treatment or of the outcome or both. We show that irrespective of the actual underlying causal structure, if any subset of the observed covariates suﬃces to control for confounding then the set of covariates chosen by our criterion will also suﬃce. We show that other, commonly used, criteria for confounding control do not have this property. We use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams. An investigator simply need ask, “Is the covariate a cause of the treatment?” and “Is the covariate a cause of the outcome?” If the answer to either question is “yes” then the covariate is included for confounder control. We discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion.},
	language = {en},
	number = {4},
	urldate = {2022-12-31},
	journal = {Biometrics},
	author = {VanderWeele, Tyler J. and Shpitser, Ilya},
	month = dec,
	year = {2011},
	pages = {1406--1413},
	file = {VanderWeele and Shpitser - 2011 - A New Criterion for Confounder Selection.pdf:/Users/jacquelinemaasch/Zotero/storage/J6Y6FH74/VanderWeele and Shpitser - 2011 - A New Criterion for Confounder Selection.pdf:application/pdf},
}

@article{vanderweele_definition_2013,
	title = {On the definition of a confounder},
	volume = {41},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-41/issue-1/On-the-definition-of-a-confounder/10.1214/12-AOS1058.full},
	doi = {10.1214/12-AOS1058},
	language = {en},
	number = {1},
	urldate = {2022-12-31},
	journal = {The Annals of Statistics},
	author = {VanderWeele, Tyler J. and Shpitser, Ilya},
	month = feb,
	year = {2013},
	file = {VanderWeele and Shpitser - 2013 - On the definition of a confounder.pdf:/Users/jacquelinemaasch/Zotero/storage/K8259LRA/VanderWeele and Shpitser - 2013 - On the definition of a confounder.pdf:application/pdf},
}

@inproceedings{kuang_stable_2018,
	address = {London United Kingdom},
	title = {Stable {Prediction} across {Unknown} {Environments}},
	isbn = {978-1-4503-5552-0},
	url = {https://dl.acm.org/doi/10.1145/3219819.3220082},
	doi = {10.1145/3219819.3220082},
	abstract = {In many important machine learning applications, the training distribution used to learn a probabilistic classifier differs from the distribution on which the classifier will be used to make predictions. Traditional methods correct the distribution shift by reweighting training data with the ratio of the density between test and training data. However, in many applications training takes place without prior knowledge of the testing distribution. Recently, methods have been proposed to address the shift by learning the underlying causal structure, but those methods rely on diversity arising from multiple training data sets, and they further have complexity limitations in high dimensions. In this paper, we propose a novel Deep Global Balancing Regression (DGBR) algorithm to jointly optimize a deep auto-encoder model for feature selection and a global balancing model for stable prediction across unknown environments. The global balancing model constructs balancing weights that facilitate estimation of partial effects of features (holding fixed all other features), a problem that is challenging in high dimensions, and thus helps to identify stable, causal relationships between features and outcomes. The deep auto-encoder model is designed to reduce the dimensionality of the feature space, thus making global balancing easier. We show, both theoretically and with empirical experiments, that our algorithm can make stable predictions across unknown environments. Our experiments on both synthetic and real datasets demonstrate that our algorithm outperforms the state-of-the-art methods for stable prediction across unknown environments.},
	language = {en},
	urldate = {2023-01-07},
	booktitle = {Proceedings of the 24th {ACM} {SIGKDD} {International} {Conference} on {Knowledge} {Discovery} \& {Data} {Mining}},
	publisher = {ACM},
	author = {Kuang, Kun and Cui, Peng and Athey, Susan and Xiong, Ruoxuan and Li, Bo},
	month = jul,
	year = {2018},
	pages = {1617--1626},
	file = {Kuang et al. - 2018 - Stable Prediction across Unknown Environments.pdf:/Users/jacquelinemaasch/Zotero/storage/64BNGQCT/Kuang et al. - 2018 - Stable Prediction across Unknown Environments.pdf:application/pdf},
}

@article{ling_bamb_2019,
	title = {{BAMB}: {A} {Balanced} {Markov} {Blanket} {Discovery} {Approach} to {Feature} {Selection}},
	volume = {10},
	issn = {2157-6904, 2157-6912},
	shorttitle = {{BAMB}},
	url = {https://dl.acm.org/doi/10.1145/3335676},
	doi = {10.1145/3335676},
	abstract = {The discovery of Markov blanket (MB) for feature selection has attracted much attention in recent years, since the MB of the class attribute is the optimal feature subset for feature selection. However, almost all existing MB discovery algorithms focus on either improving computational efficiency or boosting learning accuracy, instead of both. In this article, we propose a novel MB discovery algorithm for balancing efficiency and accuracy, called {\textless}underline{\textgreater}BA{\textless}/underline{\textgreater}lanced {\textless}underline{\textgreater}M{\textless}/underline{\textgreater}arkov {\textless}underline{\textgreater}B{\textless}/underline{\textgreater}lanket (BAMB) discovery. To achieve this goal, given a class attribute of interest, BAMB finds candidate PC (parents and children) and spouses and removes false positives from the candidate MB set in one go. Specifically, once a feature is successfully added to the current PC set, BAMB finds the spouses with regard to this feature, then uses the updated PC and the spouse set to remove false positives from the current MB set. This makes the PC and spouses of the target as small as possible and thus achieves a trade-off between computational efficiency and learning accuracy. In the experiments, we first compare BAMB with 8 state-of-the-art MB discovery algorithms on 7 benchmark Bayesian networks, then we use 10 real-world datasets and compare BAMB with 12 feature selection algorithms, including 8 state-of-the-art MB discovery algorithms and 4 other well-established feature selection methods. On prediction accuracy, BAMB outperforms 12 feature selection algorithms compared. On computational efficiency, BAMB is close to the IAMB algorithm while it is much faster than the remaining seven MB discovery algorithms.},
	language = {en},
	number = {5},
	urldate = {2023-01-09},
	journal = {ACM Transactions on Intelligent Systems and Technology},
	author = {Ling, Zhaolong and Yu, Kui and Wang, Hao and Liu, Lin and Ding, Wei and Wu, Xindong},
	month = sep,
	year = {2019},
	pages = {1--25},
	file = {Ling et al. - 2019 - BAMB A Balanced Markov Blanket Discovery Approach.pdf:/Users/jacquelinemaasch/Zotero/storage/GXRTWQDU/Ling et al. - 2019 - BAMB A Balanced Markov Blanket Discovery Approach.pdf:application/pdf},
}

@article{yu_multi-source_2020,
	title = {Multi-{Source} {Causal} {Feature} {Selection}},
	volume = {42},
	issn = {0162-8828, 2160-9292, 1939-3539},
	url = {https://ieeexplore.ieee.org/document/8677287/},
	doi = {10.1109/TPAMI.2019.2908373},
	abstract = {Causal feature selection has attracted much attention in recent years, as the causal features selected imply the causal mechanism related to the class attribute, leading to more reliable prediction models built using them. Currently there is a need of developing multi-source feature selection methods, since in many applications data for studying the same problem has been collected from various sources, such as multiple gene expression datasets obtained from different experiments for studying the causes of the same disease. However, the state-of-the-art causal feature selection methods generally tackle a single dataset, and a direct application of the methods to multiple datasets will result in unreliable results as the datasets may have different distributions. To address the challenges, by utilizing the concept of causal invariance in causal inference, we ﬁrst formulate the problem of causal feature selection with multiple datasets as a search problem for an invariant set across the datasets, then give the upper and lower bounds of the invariant set, and ﬁnally we propose a new Multi-source Causal Feature Selection algorithm, MCFS. Using synthetic and real world datasets and 16 feature selection methods, the extensive experiments have validated the effectiveness of MCFS.},
	language = {en},
	number = {9},
	urldate = {2023-01-09},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	author = {Yu, Kui and Liu, Lin and Li, Jiuyong and Ding, Wei and Le, Thuc Duy},
	month = sep,
	year = {2020},
	pages = {2240--2256},
	file = {Yu et al. - 2020 - Multi-Source Causal Feature Selection.pdf:/Users/jacquelinemaasch/Zotero/storage/PRJFYLII/Yu et al. - 2020 - Multi-Source Causal Feature Selection.pdf:application/pdf},
}

@article{wu_accurate_2020,
	title = {Accurate {Markov} {Boundary} {Discovery} for {Causal} {Feature} {Selection}},
	volume = {50},
	issn = {2168-2267, 2168-2275},
	url = {https://ieeexplore.ieee.org/document/8877773/},
	doi = {10.1109/TCYB.2019.2940509},
	abstract = {Causal feature selection has achieved much attention in recent years, which discovers a Markov boundary (MB) of the class attribute. The MB of the class attribute implies local causal relations between the class attribute and the features, thus leading to more interpretable and robust prediction models than the features selected by the traditional feature selection algorithms. Many causal feature selection methods have been proposed, and almost all of them employ conditional independence (CI) tests to identify MBs. However, many datasets from real-world applications may suffer from incorrect CI tests due to noise or small-sized samples, resulting in lower MB discovery accuracy for these existing algorithms. To tackle this issue, in this article, we ﬁrst introduce a new concept of PCMasking to explain a type of incorrect CI tests in the MB discovery, then propose a cross-check and complement MB discovery (CCMB) algorithm to repair this type of incorrect CI tests for accurate MB discovery. To improve the efﬁciency of CCMB, we further design a pipeline machine-based CCMB (PM-CCMB) algorithm. Using benchmark Bayesian network datasets, the experiments demonstrate that both CCMB and PM-CCMB achieve significant improvements on the MB discovery accuracy compared with the existing methods, and PM-CCMB further improves the computational efﬁciency. The empirical study in the real-world datasets validates the effectiveness of CCMB and PM-CCMB against the state-of-the-art causal and traditional feature selection algorithms.},
	language = {en},
	number = {12},
	urldate = {2023-01-09},
	journal = {IEEE Transactions on Cybernetics},
	author = {Wu, Xingyu and Jiang, Bingbing and Yu, Kui and Miao, chunyan and Chen, Huanhuan},
	month = dec,
	year = {2020},
	pages = {4983--4996},
	file = {Wu et al. - 2020 - Accurate Markov Boundary Discovery for Causal Feat.pdf:/Users/jacquelinemaasch/Zotero/storage/GR9AYDD8/Wu et al. - 2020 - Accurate Markov Boundary Discovery for Causal Feat.pdf:application/pdf},
}

@article{wu_multi-label_2020,
	title = {Multi-{Label} {Causal} {Feature} {Selection}},
	volume = {34},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/6114},
	doi = {10.1609/aaai.v34i04.6114},
	abstract = {Multi-label feature selection has received considerable attentions during the past decade. However, existing algorithms do not attempt to uncover the underlying causal mechanism, and individually solve different types of variable relationships, ignoring the mutual effects between them. Furthermore, these algorithms lack of interpretability, which can only select features for all labels, but cannot explain the correlation between a selected feature and a certain label. To address these problems, in this paper, we theoretically study the causal relationships in multi-label data, and propose a novel Markov blanket based multi-label causal feature selection (MB-MCF) algorithm. MB-MCF mines the causal mechanism of labels and features ﬁrst, to obtain a complete representation of information about labels. Based on the causal relationships, MBMCF then selects predictive features and simultaneously distinguishes common features shared by multiple labels and label-speciﬁc features owned by single labels. Experiments on real-world data sets validate that MB-MCF could automatically determine the number of selected features and simultaneously achieve the best performance compared with state-of-the-art methods. An experiment in Emotions data set further demonstrates the interpretability of MB-MCF.},
	language = {en},
	number = {04},
	urldate = {2023-01-09},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Wu, Xingyu and Jiang, Bingbing and Yu, Kui and Chen, Huanhuan and Miao, Chunyan},
	month = apr,
	year = {2020},
	pages = {6430--6437},
	file = {Wu et al. - 2020 - Multi-Label Causal Feature Selection.pdf:/Users/jacquelinemaasch/Zotero/storage/832HGE2E/Wu et al. - 2020 - Multi-Label Causal Feature Selection.pdf:application/pdf},
}

@article{yu_unified_2021,
	title = {A {Unified} {View} of {Causal} and {Non}-causal {Feature} {Selection}},
	volume = {15},
	issn = {1556-4681, 1556-472X},
	url = {https://dl.acm.org/doi/10.1145/3436891},
	doi = {10.1145/3436891},
	abstract = {In this article, we aim to develop a unified view of causal and non-causal feature selection methods. The unified view will fill in the gap in the research of the relation between the two types of methods. Based on the Bayesian network framework and information theory, we first show that causal and non-causal feature selection methods share the same objective. That is to find the Markov blanket of a class attribute, the theoretically optimal feature set for classification. We then examine the assumptions made by causal and non-causal feature selection methods when searching for the optimal feature set, and unify the assumptions by mapping them to the restrictions on the structure of the Bayesian network model of the studied problem. We further analyze in detail how the structural assumptions lead to the different levels of approximations employed by the methods in their search, which then result in the approximations in the feature sets found by the methods with respect to the optimal feature set. With the unified view, we can interpret the output of non-causal methods from a causal perspective and derive the error bounds of both types of methods. Finally, we present practical understanding of the relation between causal and non-causal methods using extensive experiments with synthetic data and various types of real-world data.},
	language = {en},
	number = {4},
	urldate = {2023-01-09},
	journal = {ACM Transactions on Knowledge Discovery from Data},
	author = {Yu, Kui and Liu, Lin and Li, Jiuyong},
	month = aug,
	year = {2021},
	pages = {1--46},
	file = {Yu et al. - 2021 - A Unified View of Causal and Non-causal Feature Se.pdf:/Users/jacquelinemaasch/Zotero/storage/QTT7KLYT/Yu et al. - 2021 - A Unified View of Causal and Non-causal Feature Se.pdf:application/pdf},
}

@article{yu_causal_2022,
	title = {Causal {Feature} {Selection} with {Missing} {Data}},
	volume = {16},
	issn = {1556-4681, 1556-472X},
	url = {https://dl.acm.org/doi/10.1145/3488055},
	doi = {10.1145/3488055},
	abstract = {Causal feature selection aims at learning the Markov blanket (MB) of a class variable for feature selection. The MB of a class variable implies the local causal structure among the class variable and its MB and all other features are probabilistically independent of the class variable conditioning on its MB, this enables causal feature selection to identify potential causal features for feature selection for building robust and physically meaningful prediction models. Missing data, ubiquitous in many real-world applications, remain an open research problem in causal feature selection due to its technical complexity. In this article, we discuss a novel multiple imputation MB (MimMB) framework for causal feature selection with missing data. MimMB integrates Data Imputation with MB Learning in a unified framework to enable the two key components to engage with each other. MB Learning enables Data Imputation in a potentially causal feature space for achieving accurate data imputation, while accurate Data Imputation helps MB Learning identify a reliable MB of the class variable in turn. Then, we further design an enhanced kNN estimator for imputing missing values and instantiate the MimMB. In our comprehensively experimental evaluation, our new approach can effectively learn the MB of a given variable in a Bayesian network and outperforms other rival algorithms using synthetic and real-world datasets.},
	language = {en},
	number = {4},
	urldate = {2023-01-09},
	journal = {ACM Transactions on Knowledge Discovery from Data},
	author = {Yu, Kui and Yang, Yajing and Ding, Wei},
	month = aug,
	year = {2022},
	pages = {1--24},
	file = {Yu et al. - 2022 - Causal Feature Selection with Missing Data.pdf:/Users/jacquelinemaasch/Zotero/storage/FLSS4UZX/Yu et al. - 2022 - Causal Feature Selection with Missing Data.pdf:application/pdf},
}

@article{guo_error-aware_2022,
	title = {Error-aware {Markov} blanket learning for causal feature selection},
	volume = {589},
	issn = {00200255},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0020025521013402},
	doi = {10.1016/j.ins.2021.12.118},
	abstract = {Causal feature selection has attracted much attention in recent years, since it has better robustness than the traditional feature selection. Existing causal feature selection algorithms aim to identify a Markov blanket (MB) of the class variable. The MB of the class variable implies potential local causal relations around the class variable and has been proven to be the optimal feature subset for feature selection. Since almost all existing causal feature selection methods employ conditional independence (CI) tests to learn MBs, in practical settings, existing causal feature selection algorithms encounter the problem of CI test errors, which seriously deteriorates the performance of those existing methods. To solve this issue, in this paper, we propose an Error-Aware Markov Blanket learning (EAMB) algorithm with two novel subroutines to tackle the CI test error problem. Speciﬁcally, EAMB ﬁrst identiﬁes the MB of the class variable using one subroutine, and then utilizes the other subroutine to selectively recover the missed true MB features from the discarded features. The extensive experiments on 13 real-world datasets validate the effectiveness of EAMB against fourteen state-of-the-art causal feature selection algorithms and four wellestablished traditional feature selection methods.},
	language = {en},
	urldate = {2023-01-09},
	journal = {Information Sciences},
	author = {Guo, Xianjie and Yu, Kui and Cao, Fuyuan and Li, Peipei and Wang, Hao},
	month = apr,
	year = {2022},
	pages = {849--877},
	file = {Guo et al. - 2022 - Error-aware Markov blanket learning for causal fea.pdf:/Users/jacquelinemaasch/Zotero/storage/9FTT68PV/Guo et al. - 2022 - Error-aware Markov blanket learning for causal fea.pdf:application/pdf},
}

@article{yu_causality-based_2021,
	title = {Causality-based {Feature} {Selection}: {Methods} and {Evaluations}},
	volume = {53},
	issn = {0360-0300, 1557-7341},
	shorttitle = {Causality-based {Feature} {Selection}},
	url = {https://dl.acm.org/doi/10.1145/3409382},
	doi = {10.1145/3409382},
	abstract = {Feature selection is a crucial preprocessing step in data analytics and machine learning. Classical feature selection algorithms select features based on the correlations between predictive features and the class variable and do not attempt to capture causal relationships between them. It has been shown that the knowledge about the causal relationships between features and the class variable has potential benefits for building interpretable and robust prediction models, since causal relationships imply the underlying mechanism of a system. Consequently, causality-based feature selection has gradually attracted greater attentions and many algorithms have been proposed. In this article, we present a comprehensive review of recent advances in causality-based feature selection. To facilitate the development of new algorithms in the research area and make it easy for the comparisons between new methods and existing ones, we develop the first open-source package, called CausalFS, which consists of most of the representative causality-based feature selection algorithms (available at https://github.com/kuiy/CausalFS). Using CausalFS, we conduct extensive experiments to compare the representative algorithms with both synthetic and real-world datasets. Finally, we discuss some challenging problems to be tackled in future research.},
	language = {en},
	number = {5},
	urldate = {2023-01-09},
	journal = {ACM Computing Surveys},
	author = {Yu, Kui and Guo, Xianjie and Liu, Lin and Li, Jiuyong and Wang, Hao and Ling, Zhaolong and Wu, Xindong},
	month = sep,
	year = {2021},
	pages = {1--36},
	file = {Yu et al. - 2021 - Causality-based Feature Selection Methods and Eva.pdf:/Users/jacquelinemaasch/Zotero/storage/GJGNJCST/Yu et al. - 2021 - Causality-based Feature Selection Methods and Eva.pdf:application/pdf},
}

@inproceedings{mastakouri_necessary_2021,
	title = {Necessary and {Sufficient} {Conditions} for {Causal} {Feature} {Selection} in {Time} {Series} with {Latent} {Common} {Causes}},
	booktitle = {Proceedings of the 38th {International} {Conference} on {Machine} {Learning}},
	author = {Mastakouri, Atalanti A. and Schölkopf, Bernhard and Janzing, Dominik},
	year = {2021},
	file = {Mastakouri et al. - 2021 - Necessary and Sufficient Conditions for Causal Fea.pdf:/Users/jacquelinemaasch/Zotero/storage/8MTF5VS3/Mastakouri et al. - 2021 - Necessary and Sufficient Conditions for Causal Fea.pdf:application/pdf},
}

@article{yu_learning_2020,
	title = {Learning {Markov} {Blankets} {From} {Multiple} {Interventional} {Data} {Sets}},
	volume = {31},
	issn = {2162-237X, 2162-2388},
	url = {https://ieeexplore.ieee.org/document/8818658/},
	doi = {10.1109/TNNLS.2019.2927636},
	abstract = {Learning Markov blankets (MBs) plays an important role in many machine learning tasks, such as causal Bayesian network structure learning, feature selection, and domain adaptation. Since variables included in the MB of a target variable of interest have causal relationships with the target, the MB can serve as the basis of learning the global structure of a causal Bayesian network or as a reliable and robust feature set for classiﬁcation, both within the same domain or across domains. In this article, we study the problem of learning the MB of a target variable from multiple interventional data sets. Data sets attained from interventional experiments contain richer causal information than passively observed data (observational data) for MB discovery. However, almost all existing MB discovery methods are designed for learning MBs from a single observational data set. To learn MBs from multiple interventional data sets, we face two challenges: 1) unknown intervention variables and 2) nonidentical data distributions. To address these challenges, we theoretically analyze: 1) under what conditions we can ﬁnd the correct MB of a target variable and 2) under what conditions we can identify the causes of the target variable via discovering its MB. Based on the theoretical analysis, we propose a new algorithm for learning MBs from multiple interventional data sets, and we present the conditions/assumptions that assure the correctness of the algorithm. To the best of our knowledge, this article is the ﬁrst to present the theoretical analyses about the conditions for MB discovery in multiple interventional data sets and the algorithm to ﬁnd the MBs in relation to the conditions. Using benchmark Bayesian networks and real-world data sets, the experiments have validated the effectiveness and efﬁciency of the proposed algorithm in this article.},
	language = {en},
	number = {6},
	urldate = {2023-01-09},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	author = {Yu, Kui and Liu, Lin and Li, Jiuyong},
	month = jun,
	year = {2020},
	pages = {2005--2019},
	file = {Yu et al. - 2020 - Learning Markov Blankets From Multiple Interventio.pdf:/Users/jacquelinemaasch/Zotero/storage/U6BVYIGA/Yu et al. - 2020 - Learning Markov Blankets From Multiple Interventio.pdf:application/pdf},
}

@inproceedings{claassen_greedy_2022,
	title = {Greedy {Equivalence} {Search} in the {Presence} of {Latent} {Confounders}},
	volume = {180},
	abstract = {We investigate Greedy PAG Search (GPS) for score-based causal discovery over equivalence classes, similar to the famous Greedy Equivalence Search algorithm, except now in the presence of latent confounders. It is based on a novel characterization of Markov equivalence classes for MAGs, that not only improves state-of-the-art identification of Markov equivalence between MAGs to linear time complexity for sparse graphs, but also allows for efficient traversal over equivalence classes in the space of all MAGs. The resulting GPS algorithm is evaluated against several existing alternatives and found to show promising performance, both in terms of speed and accuracy.},
	language = {en},
	booktitle = {Proceedings of the 38th {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	publisher = {PMLR},
	author = {Claassen, Tom and Bucur, Ioan Gabriel},
	year = {2022},
	file = {Claassen and Bucur - Greedy Equivalence Search in the Presence of Laten.pdf:/Users/jacquelinemaasch/Zotero/storage/NKF2EN7J/Claassen and Bucur - Greedy Equivalence Search in the Presence of Laten.pdf:application/pdf},
}

@inproceedings{chickering_statistically_2020,
	title = {Statistically {Efﬁcient} {Greedy} {Equivalence} {Search}},
	abstract = {We establish the theoretical foundation for statistically efﬁcient variants of the Greedy Equivalence Search algorithm. If each node in the generative structure has at most k parents, we show that in the limit of large data, we can recover that structure using greedy search with operator scores that condition on at most k variables. We present simple synthetic experiments that compare a backward-only variant of the new algorithm to GES using ﬁnite data, showing increasing beneﬁt of the new algorithm as the complexity of the generative model increases.},
	booktitle = {Proceedings of the 36th {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Chickering, David Maxwell},
	year = {2020},
	file = {Chickering - Statistically Efﬁcient Greedy Equivalence Search.pdf:/Users/jacquelinemaasch/Zotero/storage/YMYUEF4N/Chickering - Statistically Efﬁcient Greedy Equivalence Search.pdf:application/pdf},
}

@article{borboudakis_forward-backward_2019,
	title = {Forward-{Backward} {Selection} with {Early} {Dropping}},
	volume = {20},
	abstract = {Forward-backward selection is one of the most basic and commonly-used feature selection algorithms available. It is also general and conceptually applicable to many diﬀerent types of data. In this paper, we propose a heuristic that signiﬁcantly improves its running time, while preserving predictive performance. The idea is to temporarily discard the variables that are conditionally independent with the outcome given the selected variable set. Depending on how those variables are reconsidered and reintroduced, this heuristic gives rise to a family of algorithms with increasingly stronger theoretical guarantees. In distributions that can be faithfully represented by Bayesian networks or maximal ancestral graphs, members of this algorithmic family are able to correctly identify the Markov blanket in the sample limit. In experiments we show that the proposed heuristic increases computational eﬃciency by about 1-2 orders of magnitude, while selecting fewer or the same number of variables and retaining predictive performance. Furthermore, we show that the proposed algorithm and feature selection with LASSO perform similarly when restricted to select the same number of variables, making the proposed algorithm an attractive alternative for problems where no (eﬃcient) algorithm for LASSO exists.},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Borboudakis, Giorgos and Tsamardinos, Ioannis},
	year = {2019},
	pages = {1--39},
	file = {Borboudakis and Tsamardinos - 2019 - Forward-Backward Selection with Early Dropping.pdf:/Users/jacquelinemaasch/Zotero/storage/5LYJBYHJ/Borboudakis and Tsamardinos - 2019 - Forward-Backward Selection with Early Dropping.pdf:application/pdf},
}

@article{tsamardinos_greedy_2019,
	title = {A greedy feature selection algorithm for {Big} {Data} of high dimensionality},
	volume = {108},
	issn = {0885-6125, 1573-0565},
	url = {http://link.springer.com/10.1007/s10994-018-5748-7},
	doi = {10.1007/s10994-018-5748-7},
	abstract = {We present the Parallel, Forward–Backward with Pruning (PFBP) algorithm for feature selection (FS) for Big Data of high dimensionality. PFBP partitions the data matrix both in terms of rows as well as columns. By employing the concepts of p-values of conditional independence tests and meta-analysis techniques, PFBP relies only on computations local to a partition while minimizing communication costs, thus massively parallelizing computations. Similar techniques for combining local computations are also employed to create the ﬁnal predictive model. PFBP employs asymptotically sound heuristics to make early, approximate decisions, such as Early Dropping of features from consideration in subsequent iterations, Early Stopping of consideration of features within the same iteration, or Early Return of the winner in each iteration. PFBP provides asymptotic guarantees of optimality for data distributions faithfully representable by a causal network (Bayesian network or maximal ancestral graph). Empirical analysis conﬁrms a super-linear speedup of the algorithm with increasing sample size, linear scalability with respect to the number of features and processing cores. An extensive comparative evaluation also demonstrates the effectiveness of PFBP against other algorithms in its class. The heuristics presented are general and could potentially be employed to other greedy-type of FS algorithms. An application on simulated Single Nucleotide Polymorphism (SNP) data with 500K samples is provided as a use case.},
	language = {en},
	number = {2},
	urldate = {2023-01-16},
	journal = {Machine Learning},
	author = {Tsamardinos, Ioannis and Borboudakis, Giorgos and Katsogridakis, Pavlos and Pratikakis, Polyvios and Christophides, Vassilis},
	month = feb,
	year = {2019},
	pages = {149--202},
	file = {Tsamardinos et al. - 2019 - A greedy feature selection algorithm for Big Data .pdf:/Users/jacquelinemaasch/Zotero/storage/NBY44HTL/Tsamardinos et al. - 2019 - A greedy feature selection algorithm for Big Data .pdf:application/pdf},
}

@article{welch_algorithmic_1982,
	title = {Algorithmic complexity: {Three} {NP}-hard problems in computational statistics},
	volume = {15},
	number = {1},
	journal = {Journal of Statistical Computation and Simulation},
	author = {Welch, William J.},
	year = {1982},
	pages = {17--25},
	file = {Welch - 1982 - Algorithmic complexity Three NP-hard problems in .pdf:/Users/jacquelinemaasch/Zotero/storage/9XNS77SM/Welch - 1982 - Algorithmic complexity Three NP-hard problems in .pdf:application/pdf},
}

@article{niinimaki_local_nodate,
	title = {Local {Structure} {Discovery} in {Bayesian} {Networks}},
	abstract = {Learning a Bayesian network structure from data is an NP-hard problem and thus exact algorithms are feasible only for small data sets. Therefore, network structures for larger networks are usually learned with various heuristics. Another approach to scaling up the structure learning is local learning. In local learning, the modeler has one or more target variables that are of special interest; he wants to learn the structure near the target variables and is not interested in the rest of the variables. In this paper, we present a score-based local learning algorithm called SLL. We conjecture that our algorithm is theoretically sound in the sense that it is optimal in the limit of large sample size. Empirical results suggest that SLL is competitive when compared to the constraint-based HITON algorithm. We also study the prospects of constructing the network structure for the whole node set based on local results by presenting two algorithms and comparing them to several heuristics.},
	language = {en},
	author = {Niinimäki, Teppo and Parviainen, Pekka},
	file = {Niinimäki and Parviainen - Local Structure Discovery in Bayesian Networks.pdf:/Users/jacquelinemaasch/Zotero/storage/J6T6LMRZ/Niinimäki and Parviainen - Local Structure Discovery in Bayesian Networks.pdf:application/pdf},
}

@article{zhang_identiability_2009,
	title = {On the {Identiﬁability} of the {Post}-{Nonlinear} {Causal} {Model}},
	abstract = {By taking into account the nonlinear eﬀect of the cause, the inner noise eﬀect, and the measurement distortion eﬀect in the observed variables, the post-nonlinear (PNL) causal model has demonstrated its excellent performance in distinguishing the cause from effect. However, its identiﬁability has not been properly addressed, and how to apply it in the case of more than two variables is also a problem. In this paper, we conduct a systematic investigation on its identiﬁability in the two-variable case. We show that this model is identiﬁable in most cases; by enumerating all possible situations in which the model is not identiﬁable, we provide suﬃcient conditions for its identiﬁability. Simulations are given to support the theoretical results. Moreover, in the case of more than two variables, we show that the whole causal structure can be found by applying the PNL causal model to each structure in the Markov equivalent class and testing if the disturbance is independent of the direct causes for each variable. In this way the exhaustive search over all possible causal structures is avoided.},
	language = {en},
	journal = {Uncertainty in Artificial Intelligence},
	author = {Zhang, Kun and Hyvarinen, Aapo},
	year = {2009},
	file = {Zhang and Hyvarinen - 2009 - On the Identiﬁability of the Post-Nonlinear Causal.pdf:/Users/jacquelinemaasch/Zotero/storage/ZQN2YVB7/Zhang and Hyvarinen - 2009 - On the Identiﬁability of the Post-Nonlinear Causal.pdf:application/pdf},
}

@article{marx_weaker_nodate,
	title = {A {Weaker} {Faithfulness} {Assumption} based on {Triple} {Interactions}},
	language = {en},
	author = {Marx, Alexander and Gretton, Arthur and Mooij, Joris M},
	file = {Marx et al. - A Weaker Faithfulness Assumption based on Triple I.pdf:/Users/jacquelinemaasch/Zotero/storage/CW7YEH9S/Marx et al. - A Weaker Faithfulness Assumption based on Triple I.pdf:application/pdf},
}

@article{claassen_learning_2013,
	title = {Learning {Sparse} {Causal} {Models} is not {NP}-hard},
	abstract = {This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree k the sound and complete causal model can be obtained in worst case order N 2(k+2) independence tests, even when latent variables and selection bias may be present. We present a modiﬁcation of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks.},
	language = {en},
	journal = {Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013)},
	author = {Claassen, Tom and Mooij, Joris M and Heskes, Tom},
	year = {2013},
	file = {Claassen et al. - Learning Sparse Causal Models is not NP-hard.pdf:/Users/jacquelinemaasch/Zotero/storage/7Z7PPYIK/Claassen et al. - Learning Sparse Causal Models is not NP-hard.pdf:application/pdf},
}

@article{calderone_analysis_nodate,
	title = {Analysis of {Triplet} {Motifs} in {Biological} {Signed} {Oriented} {Graphs} {Suggests} a {Relationship} {Between} {Fine} {Topology} and {Function}},
	abstract = {Background: Networks in diﬀerent domains are characterized by similar global characteristics while diﬀering in local structures. To further extend this concept, we investigated network regularities on a ﬁne scale in order to examine the functional impact of recurring motifs in signed oriented biological networks. In this work we generalize to signaling networks some considerations made on feedback and feed forward loops and extend them by adding a close scrutiny of Linear Triplets, which have not yet been investigate in detail.},
	language = {en},
	author = {Calderone, Alberto and Cesareni, Gianni},
	file = {Calderone and Cesareni - Analysis of Triplet Motifs in Biological Signed Or.pdf:/Users/jacquelinemaasch/Zotero/storage/4KUDPW2F/Calderone and Cesareni - Analysis of Triplet Motifs in Biological Signed Or.pdf:application/pdf},
}

@article{guo_survey_2021,
	title = {A {Survey} of {Learning} {Causality} with {Data}: {Problems} and {Methods}},
	volume = {53},
	issn = {0360-0300, 1557-7341},
	shorttitle = {A {Survey} of {Learning} {Causality} with {Data}},
	url = {https://dl.acm.org/doi/10.1145/3397269},
	doi = {10.1145/3397269},
	abstract = {This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from—or the same as—the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.},
	language = {en},
	number = {4},
	urldate = {2023-02-05},
	journal = {ACM Computing Surveys},
	author = {Guo, Ruocheng and Cheng, Lu and Li, Jundong and Hahn, P. Richard and Liu, Huan},
	month = jul,
	year = {2021},
	pages = {1--37},
	file = {Guo et al. - 2021 - A Survey of Learning Causality with Data Problems.pdf:/Users/jacquelinemaasch/Zotero/storage/QHJY2BDH/Guo et al. - 2021 - A Survey of Learning Causality with Data Problems.pdf:application/pdf},
}

@article{janzing_information-geometric_2012,
	title = {Information-geometric approach to inferring causal directions},
	volume = {182-183},
	issn = {00043702},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0004370212000045},
	doi = {10.1016/j.artint.2012.01.002},
	abstract = {While conventional approaches to causal inference are mainly based on conditional (in)dependences, recent methods also account for the shape of (conditional) distributions. The idea is that the causal hypothesis “ X causes Y ” imposes that the marginal distribution P X and the conditional distribution P Y {\textbar}X represent independent mechanisms of nature. Recently it has been postulated that the shortest description of the joint distribution P X,Y should therefore be given by separate descriptions of P X and P Y {\textbar}X . Since description length in the sense of Kolmogorov complexity is uncomputable, practical implementations rely on other notions of independence. Here we deﬁne independence via orthogonality in information space. This way, we can explicitly describe the kind of dependence that occurs between P Y and P X{\textbar}Y making the causal hypothesis “Y causes X” implausible. Remarkably, this asymmetry between cause and effect becomes particularly simple if X and Y are deterministically related. We present an inference method that works in this case. We also discuss some theoretical results for the non-deterministic case although it is not clear how to employ them for a more general inference method.},
	language = {en},
	urldate = {2023-02-05},
	journal = {Artificial Intelligence},
	author = {Janzing, Dominik and Mooij, Joris and Zhang, Kun and Lemeire, Jan and Zscheischler, Jakob and Daniušis, Povilas and Steudel, Bastian and Schölkopf, Bernhard},
	month = may,
	year = {2012},
	pages = {1--31},
	file = {Janzing et al. - 2012 - Information-geometric approach to inferring causal.pdf:/Users/jacquelinemaasch/Zotero/storage/YMS8KCZZ/Janzing et al. - 2012 - Information-geometric approach to inferring causal.pdf:application/pdf},
}

@article{mooij_distinguishing_2016,
	title = {Distinguishing {Cause} from {Eﬀect} {Using} {Observational} {Data}: {Methods} and {Benchmarks}},
	volume = {17},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Mooij, Joris M and Peters, Jonas and Janzing, Dominik and Zscheischler, Jakob and Scholkopf, Bernhard},
	year = {2016},
	file = {Mooij et al. - Distinguishing Cause from Eﬀect Using Observationa.pdf:/Users/jacquelinemaasch/Zotero/storage/2CSKCRVP/Mooij et al. - Distinguishing Cause from Eﬀect Using Observationa.pdf:application/pdf},
}

@article{daniusis_inferring_2010,
	title = {Inferring deterministic causal relations},
	abstract = {We consider two variables that are related to each other by an invertible function. While it has previously been shown that the dependence structure of the noise can provide hints to determine which of the two variables is the cause, we presently show that even in the deterministic (noise-free) case, there are asymmetries that can be exploited for causal inference. Our method is based on the idea that if the function and the probability density of the cause are chosen independently, then the distribution of the eﬀect will, in a certain sense, depend on the function. We provide a theoretical analysis of this method, showing that it also works in the low noise regime, and link it to information geometry. We report strong empirical results on various real-world data sets from diﬀerent domains.},
	language = {en},
	journal = {Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence},
	author = {Daniusis, Povilas and Janzing, Dominik and Mooij, Joris and Zscheischler, Jakob and Steudel, Bastian and Zhang, Kun and Scholkopf, Bernhard},
	year = {2010},
	file = {Daniuˇsis et al. - Inferring deterministic causal relations.pdf:/Users/jacquelinemaasch/Zotero/storage/DZXBPVWK/Daniuˇsis et al. - Inferring deterministic causal relations.pdf:application/pdf},
}

@article{stegle_probabilistic_nodate,
	title = {Probabilistic latent variable models for distinguishing between cause and effect},
	abstract = {We propose a novel method for inferring whether X causes Y or vice versa from joint observations of X and Y . The basic idea is to model the observed data using probabilistic latent variable models, which incorporate the effects of unobserved noise. To this end, we consider the hypothetical effect variable to be a function of the hypothetical cause variable and an independent noise term (not necessarily additive). An important novel aspect of our work is that we do not restrict the model class, but instead put general non-parametric priors on this function and on the distribution of the cause. The causal direction can then be inferred by using standard Bayesian model selection. We evaluate our approach on synthetic data and real-world data and report encouraging results.},
	language = {en},
	author = {Stegle, Oliver and Janzing, Dominik and Zhang, Kun and Mooij, Joris M and Schölkopf, Bernhard},
	file = {Stegle et al. - Probabilistic latent variable models for distingui.pdf:/Users/jacquelinemaasch/Zotero/storage/HXSZERLS/Stegle et al. - Probabilistic latent variable models for distingui.pdf:application/pdf},
}

@article{mooij_distinguishing_2010,
	title = {Distinguishing between cause and effect},
	volume = {6},
	abstract = {We describe eight data sets that together formed the CauseEffectPairs task in the Causality Challenge \#2: Pot-Luck competition. Each set consists of a sample of a pair of statistically dependent random variables. One variable is known to cause the other one, but this information was hidden from the participants; the task was to identify which of the two variables was the cause and which one the effect, based upon the observed sample. The data sets were chosen such that we expect common agreement on the ground truth. Even though part of the statistical dependences may also be due to hidden common causes, common sense tells us that there is a signiﬁcant cause-effect relation between the two variables in each pair. We also present baseline results using three different causal inference methods.},
	language = {en},
	journal = {JMLR Workshop and Conference Proceedings},
	author = {Mooij, Joris and Janzing, Dominik},
	year = {2010},
	pages = {147--156},
	file = {Mooij and Janzing - Distinguishing between cause and effect.pdf:/Users/jacquelinemaasch/Zotero/storage/SJ454WEH/Mooij and Janzing - Distinguishing between cause and effect.pdf:application/pdf},
}

@article{chung_post-nonlinear_nodate,
	title = {Post-nonlinear {Causal} {Model} with {Deep} {Neural} {Networks}},
	abstract = {Causal discovery is a process of discovering causal information from observed data. Previous work in the ﬁeld has focused on discovering causal directions in two variables, with extra effort on constructing assumptions for the data distribution and proving identiﬁability. However they lack ﬂexibility in scaling up causal discovery for multiple variables and a systematic endto-end pipeline. We extend two existing work in the literature (LiNGAM and PNL), and by integrating them with deep neural networks, we propose an end-to-end learning scheme for training a model for causal discovery. We evaluate our method on several real and simulated datasets.},
	language = {en},
	author = {Chung, Youngseog},
	file = {Chung - Post-nonlinear Causal Model with Deep Neural Netwo.pdf:/Users/jacquelinemaasch/Zotero/storage/S83P5CG3/Chung - Post-nonlinear Causal Model with Deep Neural Netwo.pdf:application/pdf},
}

@article{achard_identifiability_2005,
	title = {Identifiability of post-nonlinear mixtures},
	volume = {12},
	issn = {1070-9908},
	url = {http://ieeexplore.ieee.org/document/1420356/},
	doi = {10.1109/LSP.2005.845593},
	abstract = {This letter deals with the resolution of the blind source separation problem using the independent component analysis method in post-nonlinear mixtures. Using the sole hypothesis of the source independence is not obvious to reconstruct the sources in nonlinear mixtures. Here, we prove the identiﬁability under weak assumptions on the mixture matrix and density sources.},
	language = {en},
	number = {5},
	urldate = {2023-02-07},
	journal = {IEEE Signal Processing Letters},
	author = {Achard, S. and Jutten, C.},
	month = may,
	year = {2005},
	pages = {423--426},
	file = {Achard and Jutten - 2005 - Identifiability of post-nonlinear mixtures.pdf:/Users/jacquelinemaasch/Zotero/storage/4Q6SMFXU/Achard and Jutten - 2005 - Identifiability of post-nonlinear mixtures.pdf:application/pdf},
}

@article{zhang_distinguishing_2008,
	title = {Distinguishing {Causes} from {Eﬀects} using {Nonlinear} {Acyclic} {Causal} {Models}},
	abstract = {Distinguishing causes from eﬀects is an important problem in many areas. In this paper, we propose a very general but well deﬁned nonlinear acyclic causal model, namely, postnonlinear acyclic causal model with inner additive noise, to tackle this problem. In this model, each observed variable is generated by a nonlinear function of its parents, with additive noise, followed by a nonlinear distortion. The nonlinearity in the second stage takes into account the eﬀect of sensor distortions, which are usually encountered in practice. In the two-variable case, if all the nonlinearities involved in the model are invertible, by relating the proposed model to the post-nonlinear independent component analysis (ICA) problem, we give the conditions under which the causal relation can be uniquely found. We present a two-step method, which is constrained nonlinear ICA followed by statistical independence tests, to distinguish the cause from the eﬀect in the two-variable case. We apply this method to solve the problem “CauseEﬀectPairs” in the Pot-luck challenge, and successfully identify causes from eﬀects.},
	language = {en},
	author = {Zhang, Kun and Hyvarinen, Aapo},
	year = {2008},
	file = {Zhang and Hyvarinen - Distinguishing Causes from Eﬀects using Nonlinear .pdf:/Users/jacquelinemaasch/Zotero/storage/JDFX2IQA/Zhang and Hyvarinen - Distinguishing Causes from Eﬀects using Nonlinear .pdf:application/pdf},
}

@inproceedings{hoyer_nonlinear_2008,
	title = {Nonlinear causal discovery with additive noise models},
	abstract = {The discovery of causal relationships between a set of observed variables is a fundamental problem in science. For continuous-valued data linear acyclic causal models with additive noise are often used because these models are well understood and there are well-known methods to ﬁt them to data. In reality, of course, many causal relationships are more or less nonlinear, raising some doubts as to the applicability and usefulness of purely linear methods. In this contribution we show that the basic linear framework can be generalized to nonlinear models. In this extended framework, nonlinearities in the data-generating process are in fact a blessing rather than a curse, as they typically provide information on the underlying causal system and allow more aspects of the true data-generating mechanisms to be identiﬁed. In addition to theoretical results we show simulations and some simple real data experiments illustrating the identiﬁcation power provided by nonlinearities.},
	language = {en},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems} 21 ({NIPS} 2008)},
	author = {Hoyer, Patrik O and Janzing, Dominik and Mooij, Joris M and Peters, Jonas and Schölkopf, Bernhard},
	year = {2008},
	file = {Hoyer et al. - Nonlinear causal discovery with additive noise mod.pdf:/Users/jacquelinemaasch/Zotero/storage/G3N2KJ6L/Hoyer et al. - Nonlinear causal discovery with additive noise mod.pdf:application/pdf},
}

@article{lemeire_causal_nodate,
	title = {Causal {Inference} on {Data} {Containing} {Deterministic} {Relations}},
	abstract = {Data containing deterministic relations cannot be handled by current constraint-based causal learning algorithms; they entail conditional independencies that cannot be represented by a faithful graph. Violation of the faithfulness property is characterized by an information equivalence of two sets of variables with respect to a reference variable. The conditional independencies do not provide information about which set should be connected to the reference variable. We propose to use the complexity of the relationships as criterion to determine adjacency. Correct decisions are made under the assumption that the complexity of relations does not increase along a causal path. This paper deﬁnes an augmented Bayesian network which explicitly models deterministic relations. The faithfulness property is redeﬁned by using a generalized deﬁnition of the d-separation criterion, which also gives the conditional independencies following from deterministic relations, and by limiting the conditional independencies that are graphically described with the simplicity condition. Based on this, an extension to the PC learning algorithm is developed that allows the construction of minimal augmented Bayesian networks from observational data. Correct models are learned from data generated by a set of structural equations.},
	language = {en},
	author = {Lemeire, Jan and Steenhaut, Kris and Maes, Sam},
	file = {Lemeire et al. - Causal Inference on Data Containing Deterministic .pdf:/Users/jacquelinemaasch/Zotero/storage/TLF66Z3U/Lemeire et al. - Causal Inference on Data Containing Deterministic .pdf:application/pdf},
}

@article{gelman_bayesian_nodate,
	title = {Bayesian {Data} {Analysis} {Third} edition (with errors ﬁxed as of 15 {February} 2021)},
	language = {en},
	author = {Gelman, Andrew and Carlin, John B and Stern, Hal S and Dunson, David B and Vehtari, Aki and Rubin, Donald B},
	file = {Gelman et al. - Bayesian Data Analysis Third edition (with errors .pdf:/Users/jacquelinemaasch/Zotero/storage/2UITA9RD/Gelman et al. - Bayesian Data Analysis Third edition (with errors .pdf:application/pdf},
}

@article{chickering_large-sample_2004,
	title = {Large-{Sample} {Learning} of {Bayesian} {Networks} is {NP}-{Hard}},
	volume = {5},
	abstract = {In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest structure for which the model is able to represent the generative distribution exactly. Our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufﬁciently large dataset. We show that identifying high-scoring structures is NPhard, even when any combination of one or more of the following hold: the generative distribution is perfect with respect to some DAG containing hidden variables; we are given an independence oracle; we are given an inference oracle; we are given an information oracle; we restrict potential solutions to structures in which each node has at most k parents, for all k ≥ 3.},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Chickering, David Maxwell and Heckerman, David and Meek, Christopher},
	year = {2004},
	pages = {1287--1330},
	file = {Chickering et al. - 2004 - Large-Sample Learning of Bayesian Networks is NP-H.pdf:/Users/jacquelinemaasch/Zotero/storage/F925DUHP/Chickering et al. - 2004 - Large-Sample Learning of Bayesian Networks is NP-H.pdf:application/pdf},
}

@article{geiger_learning_1994,
	title = {Learning {Bayesian} {Networks} in {NP}-{Hard}},
	abstract = {Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score re ecting the goodness-of- t of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et al. (1994) introduced a Bayesian metric, called the BDe metric, that computes the relative posterior probability of a network structure given data. They show that the metric has a property desireable for inferring causal structure from data. In this paper, we show that the problem of deciding whether there is a Bayesian network{\textbar}among those where each node has at most k parents{\textbar}that has a relative posterior probability greater than a given constant is NP-complete, when the BDe metric is used.},
	language = {en},
	author = {Geiger, Dan and Heckerman, David},
	year = {1994},
	file = {Geiger and Heckerman - 1994 - David M. Chickering dmax@cs.ucla.edu.pdf:/Users/jacquelinemaasch/Zotero/storage/NF9ZGNZS/Geiger and Heckerman - 1994 - David M. Chickering dmax@cs.ucla.edu.pdf:application/pdf},
}

@incollection{bickel_learning_1996,
	address = {New York, NY},
	title = {Learning {Bayesian} {Networks} is {NP}-{Complete}},
	volume = {112},
	isbn = {978-0-387-94736-5 978-1-4612-2404-4},
	url = {http://link.springer.com/10.1007/978-1-4612-2404-4_12},
	abstract = {Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score reflecting the goodness-of-fit of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et al. (1995) introduce a Bayesian metric, called the BDe metric, that computes the relative posterior probability of a network structure given data. In this paper, we show that the search problem of identifying a Bayesian network-among those where each node has at most [\{\vphantom{\}} parents-that has a relative posterior probability greater than a given constant is NP-complete, when the BDe metric is used.},
	language = {en},
	urldate = {2023-02-09},
	booktitle = {Learning from {Data}},
	publisher = {Springer New York},
	author = {Chickering, David Maxwell},
	editor = {Bickel, P. and Diggle, P. and Fienberg, S. and Krickeberg, K. and Olkin, I. and Wermuth, N. and Zeger, S. and Fisher, Doug and Lenz, Hans-J.},
	year = {1996},
	doi = {10.1007/978-1-4612-2404-4_12},
	note = {Series Title: Lecture Notes in Statistics},
	pages = {121--130},
	file = {Chickering - 1996 - Learning Bayesian Networks is NP-Complete.pdf:/Users/jacquelinemaasch/Zotero/storage/WEFB3LZD/Chickering - 1996 - Learning Bayesian Networks is NP-Complete.pdf:application/pdf},
}

@article{sudarshan_diet_nodate,
	title = {{DIET}: {Conditional} independence testing with marginal dependence measures of residual information},
	abstract = {Conditional randomization tests (crts) assess whether a variable x is predictive of another variable y, having observed covariates z. Crts require ﬁtting a large number of predictive models, which is often computationally intractable. Existing solutions to reduce the cost of crts typically split the dataset into a train and test portion, or rely on heuristics for interactions, both of which lead to a loss in power. We propose the decoupled independence test (diet), an algorithm that avoids both of these issues by leveraging marginal independence statistics to test conditional independence relationships. Diet tests the marginal independence of two random variables: F (x {\textbar} z) and F (y {\textbar} z) where F (· {\textbar} z) is a conditional cumulative distribution function (cdf). These variables are termed “information residuals.” We give suﬃcient conditions for diet to achieve ﬁnite sample type-1 error control and power greater than the type-1 error rate. We then prove that when using the mutual information between the information residuals as a test statistic, diet yields the most powerful conditionally valid test. Finally, we show diet achieves higher power than other tractable crts on several synthetic and real benchmarks.},
	language = {en},
	author = {Sudarshan, Mukund and Puli, Aahlad Manas and Tansey, Wesley and Ranganath, Rajesh},
	file = {Sudarshan et al. - DIET Conditional independence testing with margin.pdf:/Users/jacquelinemaasch/Zotero/storage/R5MK7K63/Sudarshan et al. - DIET Conditional independence testing with margin.pdf:application/pdf},
}

@article{zhang_measuring_2019,
	title = {Measuring {Conditional} {Independence} by {Independent} {Residuals} for {Causal} {Discovery}},
	volume = {10},
	issn = {2157-6904, 2157-6912},
	url = {https://dl.acm.org/doi/10.1145/3325708},
	doi = {10.1145/3325708},
	abstract = {We investigate the relationship between conditional independence (CI)
              x
              ⫫
              y
              {\textbar}
              Z
              and the independence of two residuals
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ), where
              x
              and
              y
              are two random variables and
              Z
              is a set of random variables. We show that if
              x
              ,
              y
              , and
              Z
              are generated by following linear structural equation models and all external influences follow joint Gaussian distribution, then
              x
              ⫫
              y
              {\textbar}
              Z
              if and only if
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ). That is, the test of
              x
              ⫫
              y
              {\textbar}
              Z
              can be relaxed to a simpler unconditional independence test of
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ). Furthermore, testing
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ) can be simplified by testing
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              or
              y
              −E(
              y
              {\textbar}
              Z
              )⫫
              x
              . On the other side, if all these external influences follow non-Gaussian distributions and the model satisfies structural faithfulness condition, then we have
              x
              ⫫
              y
              {\textbar}
              Z
              ⇔
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ).
            
            
              We apply the results above to the causal discovery problem, where the causal directions are generally determined by a set of
              V
              -structures and their consistent propagations, so CI test-based methods can return a set of Markov equivalence classes. We show that in the linear non-Gaussian context, in many cases
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              z
              or
              y
              −E(
              y
              {\textbar}
              Z
              )⫫
              z
              (∀
              z
              ∈
              Z
              and
              Z
              is a minimal
              d
              -separator) is satisfied when
              x
              −E(
              x
              {\textbar}
              Z
              )⫫
              y
              −E(
              y
              {\textbar}
              Z
              ), which implies
              z
              causes
              x
              (or
              y
              ) if
              z
              directly connects to
              x
              (or
              y
              ). Therefore, we conclude that CIs have useful information for distinguishing Markov equivalence classes.
            
            In summary, comparing with the existing discretization-based and kernel-based CI testing methods, the proposed method provides a simpler way to measure CI, which needs only one unconditional independence test and two regression operations. When being applied to causal discovery, it can find more causal relationships, which is extensively validated by experiments.},
	language = {en},
	number = {5},
	urldate = {2023-02-10},
	journal = {ACM Transactions on Intelligent Systems and Technology},
	author = {Zhang, Hao and Zhou, Shuigeng and Guan, Jihong and Huan, Jun (Luke)},
	month = sep,
	year = {2019},
	pages = {1--19},
	file = {Zhang et al. - 2019 - Measuring Conditional Independence by Independent .pdf:/Users/jacquelinemaasch/Zotero/storage/F27F4CZV/Zhang et al. - 2019 - Measuring Conditional Independence by Independent .pdf:application/pdf},
}

@article{katsevich_power_2022,
	title = {On the power of conditional independence testing under model-{X}*},
	volume = {16},
	issn = {1935-7524},
	url = {https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-16/issue-2/On-the-power-of-conditional-independence-testing-under-model-X/10.1214/22-EJS2085.full},
	doi = {10.1214/22-EJS2085},
	abstract = {For testing conditional independence (CI) of a response Y and a predictor X given covariates Z, the model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoﬀs and their application to genome-wide association studies. In this paper, we study the power of MX CI tests, yielding quantitative insights into the role of machine learning and providing evidence in favor of using likelihood-based statistics in practice. Focusing on the conditional randomization test (CRT), we ﬁnd that its conditional mode of inference allows us to reformulate it as testing a point null hypothesis involving the conditional distribution of X. The Neyman-Pearson lemma implies that a likelihoodbased statistic yields the most powerful CRT against a point alternative. We obtain a related optimality result for MX knockoﬀs. Switching to an asymptotic framework with arbitrarily growing covariate dimension, we derive an expression for the power of the CRT against local semiparametric alternatives in terms of the prediction error of the machine learning algorithm on which its test statistic is based. Finally, we exhibit a resampling-free test with uniform asymptotic Type-I error control under the assumption that only the ﬁrst two moments of X given Z are known.},
	language = {en},
	number = {2},
	urldate = {2023-02-10},
	journal = {Electronic Journal of Statistics},
	author = {Katsevich, Eugene and Ramdas, Aaditya},
	month = jan,
	year = {2022},
	file = {Katsevich and Ramdas - 2022 - On the power of conditional independence testing u.pdf:/Users/jacquelinemaasch/Zotero/storage/TZLBZNMT/Katsevich and Ramdas - 2022 - On the power of conditional independence testing u.pdf:application/pdf},
}

@article{zhang_measuring_2018,
	title = {Measuring {Conditional} {Independence} by {Independent} {Residuals}: {Theoretical} {Results} and {Application} in {Causal} {Discovery}},
	volume = {32},
	issn = {2374-3468, 2159-5399},
	shorttitle = {Measuring {Conditional} {Independence} by {Independent} {Residuals}},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/11555},
	doi = {10.1609/aaai.v32i1.11555},
	abstract = {We investigate the relationship between conditional independence (CI) x ⊥ y{\textbar}Z and the independence of two residuals x − E(x{\textbar}Z) ⊥ y − E(y{\textbar}Z), where x and y are two random variables, and Z is a set of random variables. We show that if x, y and Z are generated by following linear structural equation model and all external inﬂuences follow Gaussian distributions, then x ⊥ y{\textbar}Z if and only if x − E(x{\textbar}Z) ⊥ y − E(y{\textbar}Z). That is, the test of x ⊥ y{\textbar}Z can be relaxed to a simpler unconditional independence test of x − E(x{\textbar}Z) ⊥ y − E(y{\textbar}Z). Furthermore, if all these external inﬂuences follow non-Gaussian distributions and the model satisﬁes structural faithfulness condition, then we have x ⊥ y{\textbar}Z ⇔ x − E(x{\textbar}Z) ⊥ y − E(y{\textbar}Z).},
	language = {en},
	number = {1},
	urldate = {2023-02-10},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Zhang, Hao and Zhou, Shuigeng and Guan, Jihong},
	month = apr,
	year = {2018},
	file = {Zhang et al. - 2018 - Measuring Conditional Independence by Independent .pdf:/Users/jacquelinemaasch/Zotero/storage/G2AYC23A/Zhang et al. - 2018 - Measuring Conditional Independence by Independent .pdf:application/pdf},
}

@article{zhang_residual_2022,
	title = {Residual {Similarity} {Based} {Conditional} {Independence} {Test} and {Its} {Application} in {Causal} {Discovery}},
	volume = {36},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/20539},
	doi = {10.1609/aaai.v36i5.20539},
	abstract = {Recently, many regression based conditional independence (CI) test methods have been proposed to solve the problem of causal discovery. These methods provide alternatives to test CI by ﬁrst removing the information of the controlling set from the two target variables, and then testing the independence between the corresponding residuals Res1 and Res2. When the residuals are linearly uncorrelated, the independence test between them is nontrivial. With the ability to calculate inner product in high-dimensional space, kernel-based methods are usually used to achieve this goal, but still consume considerable time. In this paper, we investigate the independence between two linear combinations under linear non-Gaussian structural equation model. We show that the dependence between the two residuals can be captured by the diﬀerence between the similarity of (Res1, Res2) and that of (Res1, Res3) (Res3 is generated by random permutation) in highdimensional space. With this result, we design a new method called SCIT for CI test, where permutation test is performed to control Type I error rate. The proposed method is simpler yet more eﬃcient and eﬀective than the existing ones. When applied to causal discovery, the proposed method outperforms the counterparts in terms of both speed and Type II error rate, especially in the case of small sample size, which is validated by our extensive experiments on various datasets.},
	language = {en},
	number = {5},
	urldate = {2023-02-10},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Zhang, Hao and Zhou, Shuigeng and Zhang, Kun and Guan, Jihong},
	month = jun,
	year = {2022},
	pages = {5942--5949},
	file = {Zhang et al. - 2022 - Residual Similarity Based Conditional Independence.pdf:/Users/jacquelinemaasch/Zotero/storage/4F5HGIL9/Zhang et al. - 2022 - Residual Similarity Based Conditional Independence.pdf:application/pdf},
}

@article{shah_hardness_2020,
	title = {The hardness of conditional independence testing and the generalised covariance measure},
	volume = {48},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-48/issue-3/The-hardness-of-conditional-independence-testing-and-the-generalised-covariance/10.1214/19-AOS1857.full},
	doi = {10.1214/19-AOS1857},
	abstract = {It is a common saying that testing for conditional independence, i.e., testing whether whether two random vectors X and Y are independent, given Z, is a hard statistical problem if Z is a continuous random variable (or vector). In this paper, we prove that conditional independence is indeed a particularly diﬃcult hypothesis to test for. Valid statistical tests are required to have a size that is smaller than a predeﬁned signiﬁcance level, and diﬀerent tests usually have power against a diﬀerent class of alternatives. We prove that a valid test for conditional independence does not have power against any alternative.},
	language = {en},
	number = {3},
	urldate = {2023-02-10},
	journal = {The Annals of Statistics},
	author = {Shah, Rajen D. and Peters, Jonas},
	month = jun,
	year = {2020},
	file = {Shah and Peters - 2020 - The hardness of conditional independence testing a.pdf:/Users/jacquelinemaasch/Zotero/storage/NIQ65RH9/Shah and Peters - 2020 - The hardness of conditional independence testing a.pdf:application/pdf},
}

@article{guo_testing_2020,
	title = {On {Testing} {Marginal} versus {Conditional} {Independence}},
	volume = {107},
	issn = {0006-3444, 1464-3510},
	url = {http://arxiv.org/abs/1906.01850},
	doi = {10.1093/biomet/asaa040},
	abstract = {We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection. To deal with this non-uniformity, we study a class of "envelope" distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are well-behaved and lead to model selection procedures with rate-free uniform error guarantees and near-optimal power. To control the error even when the two models are indistinguishable, rather than insist on a dichotomous choice, the proposed procedure will choose either or both models.},
	language = {en},
	number = {4},
	urldate = {2023-02-15},
	journal = {Biometrika},
	author = {Guo, F. Richard and Richardson, Thomas S.},
	month = dec,
	year = {2020},
	note = {arXiv:1906.01850 [math, stat]},
	keywords = {Statistics - Methodology, Mathematics - Statistics Theory},
	pages = {771--790},
	file = {Guo and Richardson - 2020 - On Testing Marginal versus Conditional Independenc.pdf:/Users/jacquelinemaasch/Zotero/storage/A4WRY9BG/Guo and Richardson - 2020 - On Testing Marginal versus Conditional Independenc.pdf:application/pdf},
}

@misc{gelman_bayesian_2020,
	title = {Bayesian {Workflow}},
	url = {http://arxiv.org/abs/2011.01808},
	abstract = {The Bayesian approach to data analysis provides a powerful way to handle uncertainty in all observations, model parameters, and model structure using probability theory. Probabilistic programming languages make it easier to specify and ﬁt Bayesian models, but this still leaves us with many options regarding constructing, evaluating, and using these models, along with many remaining challenges in computation. Using Bayesian inference to solve real-world problems requires not only statistical skills, subject matter knowledge, and programming, but also awareness of the decisions made in the process of data analysis. All of these aspects can be understood as part of a tangled workﬂow of applied Bayesian statistics. Beyond inference, the workﬂow also includes iterative model building, model checking, validation and troubleshooting of computational problems, model understanding, and model comparison. We review all these aspects of workﬂow in the context of several examples, keeping in mind that in practice we will be ﬁtting many models for any given problem, even if only a subset of them will ultimately be relevant for our conclusions.},
	language = {en},
	urldate = {2023-02-15},
	publisher = {arXiv},
	author = {Gelman, Andrew and Vehtari, Aki and Simpson, Daniel and Margossian, Charles C. and Carpenter, Bob and Yao, Yuling and Kennedy, Lauren and Gabry, Jonah and Bürkner, Paul-Christian and Modrák, Martin},
	month = nov,
	year = {2020},
	note = {arXiv:2011.01808 [stat]},
	keywords = {Statistics - Methodology},
	file = {Gelman et al. - 2020 - Bayesian Workflow.pdf:/Users/jacquelinemaasch/Zotero/storage/B6BNXRG5/Gelman et al. - 2020 - Bayesian Workflow.pdf:application/pdf},
}

@article{shimizu_linear_2006,
	title = {A {Linear} {Non}-{Gaussian} {Acyclic} {Model} for {Causal} {Discovery}},
	volume = {7},
	abstract = {In recent years, several methods have been proposed for the discovery of causal structure from non-experimental data. Such methods make various assumptions on the data generating process to facilitate its identiﬁcation from purely observational data. Continuing this line of research, we show how to discover the complete causal structure of continuous-valued data, under the assumptions that (a) the data generating process is linear, (b) there are no unobserved confounders, and (c) disturbance variables have non-Gaussian distributions of non-zero variances. The solution relies on the use of the statistical method known as independent component analysis, and does not require any pre-speciﬁed time-ordering of the variables. We provide a complete Matlab package for performing this LiNGAM analysis (short for Linear Non-Gaussian Acyclic Model), and demonstrate the effectiveness of the method using artiﬁcially generated data and real-world data.},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Shimizu, Shohei and Hoyer, Patrik O and Hyvarinen, Aapo and Kerminen, Antti},
	year = {2006},
	pages = {2003--2030},
	file = {Shimizu et al. - 2006 - A Linear Non-Gaussian Acyclic Model for Causal Dis.pdf:/Users/jacquelinemaasch/Zotero/storage/S85RSDGA/Shimizu et al. - 2006 - A Linear Non-Gaussian Acyclic Model for Causal Dis.pdf:application/pdf},
}

@inproceedings{kadowaki_estimation_2013,
	address = {Southampton, United Kingdom},
	title = {Estimation of causal structures in longitudinal data using non-{Gaussianity}},
	isbn = {978-1-4799-1180-6},
	url = {http://ieeexplore.ieee.org/document/6661912/},
	doi = {10.1109/MLSP.2013.6661912},
	language = {en},
	urldate = {2023-02-24},
	booktitle = {2013 {IEEE} {International} {Workshop} on {Machine} {Learning} for {Signal} {Processing} ({MLSP})},
	publisher = {IEEE},
	author = {Kadowaki, Kento and Shimizu, Shohei and Washio, Takashi},
	month = sep,
	year = {2013},
	pages = {1--6},
	file = {Kadowaki et al. - 2013 - Estimation of causal structures in longitudinal da.pdf:/Users/jacquelinemaasch/Zotero/storage/82R9JWHR/Kadowaki et al. - 2013 - Estimation of causal structures in longitudinal da.pdf:application/pdf},
}

@article{hyvarinen_pairwise_nodate,
	title = {Pairwise {Likelihood} {Ratios} for {Estimation} of {Non}-{Gaussian} {Structural} {Equation} {Models}},
	abstract = {We present new measures of the causal direction, or direction of effect, between two non-Gaussian random variables. They are based on the likelihood ratio under the linear non-Gaussian acyclic model (LiNGAM). We also develop simple ﬁrst-order approximations of the likelihood ratio and analyze them based on related cumulant-based measures, which can be shown to ﬁnd the correct causal directions. We show how to apply these measures to estimate LiNGAM for more than two variables, and even in the case of more variables than observations. We further extend the method to cyclic and nonlinear models. The proposed framework is statistically at least as good as existing ones in the cases of few data points or noisy data, and it is computationally and conceptually very simple. Results on simulated fMRI data indicate that the method may be useful in neuroimaging where the number of time points is typically quite small.},
	language = {en},
	author = {Hyvarinen, Aapo and Smith, Stephen M},
	file = {Hyvarinen and Smith - Pairwise Likelihood Ratios for Estimation of Non-G.pdf:/Users/jacquelinemaasch/Zotero/storage/BNG33M2K/Hyvarinen and Smith - Pairwise Likelihood Ratios for Estimation of Non-G.pdf:application/pdf},
}

@article{shimizu2011directlingam,
  title={DirectLiNGAM: A direct method for learning a linear non-Gaussian structural equation model},
  author={Shimizu, Shohei and Inazumi, Takanori and Sogawa, Yasuhiro and Hyvarinen, Aapo and Kawahara, Yoshinobu and Washio, Takashi and Hoyer, Patrik O and Bollen, Kenneth and Hoyer, Patrik},
  journal={Journal of Machine Learning Research-JMLR},
  volume={12},
  number={Apr},
  pages={1225--1248},
  year={2011}
}

@article{ikeuchi2023python,
  title={Python package for causal discovery based on LiNGAM},
  author={Ikeuchi, Takashi and Ide, Mayumi and Zeng, Yan and Maeda, Takashi Nicholas and Shimizu, Shohei},
  journal={Journal of Machine Learning Research},
  volume={24},
  number={14},
  pages={1--8},
  year={2023}
}


@inproceedings{dai2024local,
  title={Local Causal Discovery with Linear non-Gaussian Cyclic Models},
  author={Dai, Haoyue and Ng, Ignavier and Zheng, Yujia and Gao, Zhengqing and Zhang, Kun},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={154--162},
  year={2024},
  organization={PMLR}
}


@article{hyttinen_learning_2012,
	title = {Learning Linear Cyclic Causal Models with Latent Variables},
	volume = {13},
	pages = {3387--3439},
	journaltitle = {Journal of Machine Learning Research},
	author = {Hyttinen, Antti and Eberhardt, Frederick and Hoyer, Patrik O},
	langid = {english},
	year = {2012},
	file = {Hyttinen et al. - Learning Linear Cyclic Causal Models with Latent V.pdf:/Users/jmaasch/Zotero/storage/N7TGKHSY/Hyttinen et al. - Learning Linear Cyclic Causal Models with Latent V.pdf:application/pdf},
}


@inproceedings{lacerda_discovering_2008,
	title = {Discovering Cyclic Causal Models by Independent Components Analysis},
	pages = {366--374},
	booktitle = {Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence},
	author = {Lacerda, Gustavo and Spirtes, Peter and Ramsey, Joseph and Hoyer, Patrik O},
	date = {2008},
	langid = {english},
	file = {Lacerda et al. - Discovering Cyclic Causal Models by Independent Co.pdf:/Users/jmaasch/Zotero/storage/SDA69QBU/Lacerda et al. - Discovering Cyclic Causal Models by Independent Co.pdf:application/pdf},
}


@article{salehkaleybar2020learning,
  title={Learning linear non-gaussian causal models in the presence of latent variables},
  author={Salehkaleybar, Saber and Ghassami, AmirEmad and Kiyavash, Negar and Zhang, Kun},
  journal={Journal of Machine Learning Research},
  volume={21},
  number={39},
  pages={1--24},
  year={2020}
}

@article{genin2021statistical,
  title={Statistical undecidability in linear, non-gaussian causal models in the presence of latent confounders},
  author={Genin, Konstantin},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={13564--13574},
  year={2021}
}

@article{hyvarinen2013pairwise,
  title={Pairwise likelihood ratios for estimation of non-Gaussian structural equation models},
  author={Hyv{\"a}rinen, Aapo and Smith, Stephen M},
  journal={The Journal of Machine Learning Research},
  volume={14},
  number={1},
  pages={111--152},
  year={2013},
  publisher={JMLR. org}
}


@inproceedings{hoyer_bayesian_2009,
	title = {Bayesian Discovery of Linear Acyclic Causal Models},
	booktitle = {Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence},
	author = {Hoyer, Patrik O and Hyttinen, Antti},
	year = {2009},
	langid = {english},
	file = {Hoyer and Hyttinen - Bayesian Discovery of Linear Acyclic Causal Models.pdf:/Users/jmaasch/Zotero/storage/H6G9MHS5/Hoyer and Hyttinen - Bayesian Discovery of Linear Acyclic Causal Models.pdf:application/pdf},
}

@article{hyvarinen2010estimation,
  title={Estimation of a structural vector autoregression model using non-gaussianity.},
  author={Hyv{\"a}rinen, Aapo and Zhang, Kun and Shimizu, Shohei and Hoyer, Patrik O},
  journal={Journal of Machine Learning Research},
  volume={11},
  number={5},
  year={2010}
}


@article{hyvarinen_estimation_nodate,
	title = {Estimation of a {Structural} {Vector} {Autoregression} {Model} {Using} {Non}-{Gaussianity}},
	abstract = {Analysis of causal effects between continuous-valued variables typically uses either autoregressive models or structural equation models with instantaneous effects. Estimation of Gaussian, linear structural equation models poses serious identiﬁability problems, which is why it was recently proposed to use non-Gaussian models. Here, we show how to combine the non-Gaussian instantaneous model with autoregressive models. This is effectively what is called a structural vector autoregression (SVAR) model, and thus our work contributes to the long-standing problem of how to estimate SVAR’s. We show that such a non-Gaussian model is identiﬁable without prior knowledge of network structure. We propose computationally efﬁcient methods for estimating the model, as well as methods to assess the signiﬁcance of the causal inﬂuences. The model is successfully applied on ﬁnancial and brain imaging data.},
	language = {en},
	author = {Hyvarinen, Aapo and Zhang, Kun and Shimizu, Shohei and Hoyer, Patrik O},
	file = {Hyvarinen et al. - Estimation of a Structural Vector Autoregression M.pdf:/Users/jacquelinemaasch/Zotero/storage/LZ5MLL4J/Hyvarinen et al. - Estimation of a Structural Vector Autoregression M.pdf:application/pdf},
}

@incollection{wiedermann_nonlinear_2016,
	edition = {1},
	title = {Nonlinear {Functional} {Causal} {Models} for {Distinguishing} {Cause} from {Effect}},
	isbn = {978-1-118-94704-3 978-1-118-94707-4},
	url = {https://onlinelibrary.wiley.com/doi/10.1002/9781118947074.ch8},
	language = {en},
	urldate = {2023-02-24},
	booktitle = {Wiley {Series} in {Probability} and {Statistics}},
	publisher = {Wiley},
	author = {Zhang, Kun and Hyvärinen, Aapo},
	editor = {Wiedermann, Wolfgang and Eye, Alexander},
	month = jun,
	year = {2016},
	doi = {10.1002/9781118947074.ch8},
	pages = {185--201},
	file = {Zhang and Hyvärinen - 2016 - Nonlinear Functional Causal Models for Distinguish.pdf:/Users/jacquelinemaasch/Zotero/storage/3VFSQMWI/Zhang and Hyvärinen - 2016 - Nonlinear Functional Causal Models for Distinguish.pdf:application/pdf},
}

@article{kawahara_analyzing_2011,
	title = {Analyzing relationships among {ARMA} processes based on non-{Gaussianity} of external influences},
	volume = {74},
	issn = {09252312},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0925231211001147},
	doi = {10.1016/j.neucom.2011.02.008},
	abstract = {The analysis of a relationship among variables in data generating systems is one of the important problems in machine learning. In this paper, we propose an approach for estimating a graphical representation of variables in data generating processes, based on the non-Gaussianity of external inﬂuences and an autoregressive moving-average (ARMA) model. The presented model consists of two parts, i.e., a classical structural-equation model for instantaneous effects and an ARMA model for lagged effects in processes, and is estimated through the analysis using the non-Gaussianity on the residual processes. As well as the recently proposed non-Gaussianity based method named LiNGAM analysis, the estimation by the proposed method has identiﬁability and consistency. We also address the relation of the estimated structure by our method to the Granger causality. Finally, we demonstrate analyses on the data containing both of the instantaneous causality and the Granger (temporal) causality by using our proposed method where the datasets for the demonstration cover both artiﬁcial and real physical systems.},
	language = {en},
	number = {12-13},
	urldate = {2023-02-24},
	journal = {Neurocomputing},
	author = {Kawahara, Yoshinobu and Shimizu, Shohei and Washio, Takashi},
	month = jun,
	year = {2011},
	pages = {2212--2221},
	file = {Kawahara et al. - 2011 - Analyzing relationships among ARMA processes based.pdf:/Users/jacquelinemaasch/Zotero/storage/DS235CYA/Kawahara et al. - 2011 - Analyzing relationships among ARMA processes based.pdf:application/pdf},
}

@article{tashiro_parcelingam_2014,
	title = {{ParceLiNGAM}: {A} {Causal} {Ordering} {Method} {Robust} {Against} {Latent} {Confounders}},
	volume = {26},
	issn = {0899-7667, 1530-888X},
	shorttitle = {{ParceLiNGAM}},
	url = {https://direct.mit.edu/neco/article/26/1/57-83/7935},
	doi = {10.1162/NECO_a_00533},
	abstract = {We consider learning a causal ordering of variables in a linear nongaussian acyclic model called LiNGAM. Several methods have been shown to consistently estimate a causal ordering assuming that all the model assumptions are correct. But the estimation results could be distorted if some assumptions are violated. In this letter, we propose a new algorithm for learning causal orders that is robust against one typical violation of the model assumptions: latent confounders. The key idea is to detect latent confounders by testing independence between estimated external influences and find subsets (parcels) that include variables unaffected by latent confounders. We demonstrate the effectiveness of our method using artificial data and simulated brain imaging data.},
	language = {en},
	number = {1},
	urldate = {2023-02-24},
	journal = {Neural Computation},
	author = {Tashiro, Tatsuya and Shimizu, Shohei and Hyvärinen, Aapo and Washio, Takashi},
	month = jan,
	year = {2014},
	pages = {57--83},
	file = {Tashiro et al. - 2014 - ParceLiNGAM A Causal Ordering Method Robust Again.pdf:/Users/jacquelinemaasch/Zotero/storage/3NWWNY97/Tashiro et al. - 2014 - ParceLiNGAM A Causal Ordering Method Robust Again.pdf:application/pdf},
}

@article{shimizu_joint_2012,
	title = {Joint estimation of linear non-{Gaussian} acyclic models},
	volume = {81},
	issn = {09252312},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0925231211006813},
	doi = {10.1016/j.neucom.2011.11.005},
	abstract = {A linear non-Gaussian structural equation model called LiNGAM is an identiﬁable model for exploratory causal analysis. Previous methods estimate a causal ordering of variables and their connection strengths based on a single dataset. However, in many application domains, data are obtained under different conditions, that is, multiple datasets are obtained rather than a single dataset. In this paper, we present a new method to jointly estimate multiple LiNGAMs under the assumption that the models share a causal ordering but may have different connection strengths and differently distributed variables. In simulations, the new method estimates the models more accurately than estimating them separately.},
	language = {en},
	urldate = {2023-02-24},
	journal = {Neurocomputing},
	author = {Shimizu, Shohei},
	month = apr,
	year = {2012},
	pages = {104--107},
	file = {Shimizu - 2012 - Joint estimation of linear non-Gaussian acyclic mo.pdf:/Users/jacquelinemaasch/Zotero/storage/BG9MSZ8P/Shimizu - 2012 - Joint estimation of linear non-Gaussian acyclic mo.pdf:application/pdf},
}

@article{hoyer_estimation_2008,
	title = {Estimation of causal effects using linear non-{Gaussian} causal models with hidden variables},
	volume = {49},
	issn = {0888613X},
	doi = {10.1016/j.ijar.2008.02.006},
	abstract = {The task of estimating causal eﬀects from non-experimental data is notoriously diﬃcult and unreliable. Nevertheless, precisely such estimates are commonly required in many ﬁelds including economics and social science, where controlled experiments are often impossible. Linear causal models (structural equation models), combined with an implicit normality (Gaussianity) assumption on the data, provide a widely used framework for this task.},
	language = {en},
	number = {2},
	urldate = {2023-02-24},
	journal = {International Journal of Approximate Reasoning},
	author = {Hoyer, Patrik O. and Shimizu, Shohei and Kerminen, Antti J. and Palviainen, Markus},
	month = oct,
	year = {2008},
	pages = {362--378},
	file = {Hoyer et al. - 2008 - Estimation of causal effects using linear non-Gaus.pdf:/Users/jacquelinemaasch/Zotero/storage/4XN7EGHF/Hoyer et al. - 2008 - Estimation of causal effects using linear non-Gaus.pdf:application/pdf},
}

@book{wiedermann_direction_2020,
	edition = {1},
	title = {Direction {Dependence} in {Statistical} {Modeling}: {Methods} of {Analysis}},
	isbn = {978-1-119-52302-4},
	shorttitle = {Direction {Dependence} in {Statistical} {Modeling}},
	url = {https://onlinelibrary.wiley.com/doi/book/10.1002/9781119523024},
	language = {en},
	urldate = {2023-02-24},
	publisher = {Wiley},
	author = {Wiedermann, Wolfgang and Kim, Daeyoung and Sungur, Engin A. and von Eye, Alexander},
	month = dec,
	year = {2020},
	doi = {10.1002/9781119523024},
	file = {Wiedermann et al. - 2020 - Direction Dependence in Statistical Modeling Meth.pdf:/Users/jacquelinemaasch/Zotero/storage/P2HDHHI2/Wiedermann et al. - 2020 - Direction Dependence in Statistical Modeling Meth.pdf:application/pdf},
}

@article{shimizu_lingam_2014,
	title = {Lingam: {Non}-{Gaussian} {Methods} for {Estimating} {Causal} {Structures}},
	volume = {41},
	issn = {0385-7417, 1349-6964},
	shorttitle = {Lingam},
	url = {http://link.springer.com/10.2333/bhmk.41.65},
	doi = {10.2333/bhmk.41.65},
	language = {en},
	number = {1},
	urldate = {2023-02-24},
	journal = {Behaviormetrika},
	author = {Shimizu, Shohei},
	month = jan,
	year = {2014},
	pages = {65--98},
	file = {Shimizu - 2014 - Lingam Non-Gaussian Methods for Estimating Causal.pdf:/Users/jacquelinemaasch/Zotero/storage/ISRPWKB2/Shimizu - 2014 - Lingam Non-Gaussian Methods for Estimating Causal.pdf:application/pdf},
}

@article{upadhyaya_scalable_2023,
	title = {Scalable {Causal} {Structure} {Learning}: {Scoping} {Review} of {Traditional} and {Deep} {Learning} {Algorithms} and {New} {Opportunities} in {Biomedicine}},
	volume = {11},
	issn = {2291-9694},
	shorttitle = {Scalable {Causal} {Structure} {Learning}},
	url = {https://medinform.jmir.org/2023/1/e38266},
	doi = {10.2196/38266},
	abstract = {Background: Causal structure learning refers to a process of identifying causal structures from observational data, and it can have multiple applications in biomedicine and health care.
Objective: This paper provides a practical review and tutorial on scalable causal structure learning models with examples of real-world data to help health care audiences understand and apply them.
Methods: We reviewed traditional (combinatorial and score-based) methods for causal structure discovery and machine learning–based schemes. Various traditional approaches have been studied to tackle this problem, the most important among these being the Peter Spirtes and Clark Glymour algorithms. This was followed by analyzing the literature on score-based methods, which are computationally faster. Owing to the continuous constraint on acyclicity, there are new deep learning approaches to the problem in addition to traditional and score-based methods. Such methods can also offer scalability, particularly when there is a large amount of data involving multiple variables. Using our own evaluation metrics and experiments on linear, nonlinear, and benchmark Sachs data, we aimed to highlight the various advantages and disadvantages associated with these methods for the health care community. We also highlighted recent developments in biomedicine where causal structure learning can be applied to discover structures such as gene networks, brain connectivity networks, and those in cancer epidemiology.
Results: We also compared the performance of traditional and machine learning–based algorithms for causal discovery over some benchmark data sets. Directed Acyclic Graph-Graph Neural Network has the lowest structural hamming distance (19) and false positive rate (0.13) based on the Sachs data set, whereas Greedy Equivalence Search and Max-Min Hill Climbing have the best false discovery rate (0.68) and true positive rate (0.56), respectively.
Conclusions: Machine learning–based approaches, including deep learning, have many advantages over traditional approaches, such as scalability, including a greater number of variables, and potentially being applied in a wide range of biomedical applications, such as genetics, if sufficient data are available. Furthermore, these models are more flexible than traditional models and are poised to positively affect many applications in the future.},
	language = {en},
	urldate = {2023-02-27},
	journal = {JMIR Medical Informatics},
	author = {Upadhyaya, Pulakesh and Zhang, Kai and Li, Can and Jiang, Xiaoqian and Kim, Yejin},
	month = jan,
	year = {2023},
	pages = {e38266},
	file = {Upadhyaya et al. - 2023 - Scalable Causal Structure Learning Scoping Review.pdf:/Users/jacquelinemaasch/Zotero/storage/SDYUZNAW/Upadhyaya et al. - 2023 - Scalable Causal Structure Learning Scoping Review.pdf:application/pdf},
}

@article{ohlsson_applying_2020,
	title = {Applying {Causal} {Inference} {Methods} in {Psychiatric} {Epidemiology}: {A} {Review}},
	volume = {77},
	issn = {2168-622X},
	shorttitle = {Applying {Causal} {Inference} {Methods} in {Psychiatric} {Epidemiology}},
	url = {https://jamanetwork.com/journals/jamapsychiatry/fullarticle/2757020},
	doi = {10.1001/jamapsychiatry.2019.3758},
	language = {en},
	number = {6},
	urldate = {2023-02-27},
	journal = {JAMA Psychiatry},
	author = {Ohlsson, Henrik and Kendler, Kenneth S.},
	month = jun,
	year = {2020},
	pages = {637},
	file = {Ohlsson and Kendler - 2020 - Applying Causal Inference Methods in Psychiatric E.pdf:/Users/jacquelinemaasch/Zotero/storage/BFLWL46Q/Ohlsson and Kendler - 2020 - Applying Causal Inference Methods in Psychiatric E.pdf:application/pdf},
}

@article{richens_improving_2020,
	title = {Improving the accuracy of medical diagnosis with causal machine learning},
	volume = {11},
	issn = {2041-1723},
	url = {https://www.nature.com/articles/s41467-020-17419-7},
	doi = {10.1038/s41467-020-17419-7},
	abstract = {Abstract
            Machine learning promises to revolutionize clinical decision making and diagnosis. In medical diagnosis a doctor aims to explain a patient’s symptoms by determining the diseases causing them. However, existing machine learning approaches to diagnosis are purely associative, identifying diseases that are strongly correlated with a patients symptoms. We show that this inability to disentangle correlation from causation can result in sub-optimal or dangerous diagnoses. To overcome this, we reformulate diagnosis as a counterfactual inference task and derive counterfactual diagnostic algorithms. We compare our counterfactual algorithms to the standard associative algorithm and 44 doctors using a test set of clinical vignettes. While the associative algorithm achieves an accuracy placing in the top 48\% of doctors in our cohort, our counterfactual algorithm places in the top 25\% of doctors, achieving expert clinical accuracy. Our results show that causal reasoning is a vital missing ingredient for applying machine learning to medical diagnosis.},
	language = {en},
	number = {1},
	urldate = {2023-02-27},
	journal = {Nature Communications},
	author = {Richens, Jonathan G. and Lee, Ciarán M. and Johri, Saurabh},
	month = aug,
	year = {2020},
	pages = {3923},
	file = {Richens et al. - 2020 - Improving the accuracy of medical diagnosis with c.pdf:/Users/jacquelinemaasch/Zotero/storage/VJDIXLXA/Richens et al. - 2020 - Improving the accuracy of medical diagnosis with c.pdf:application/pdf},
}

@article{sanchez_causal_2022,
	title = {Causal machine learning for healthcare and precision medicine},
	volume = {9},
	issn = {2054-5703},
	url = {https://royalsocietypublishing.org/doi/10.1098/rsos.220638},
	doi = {10.1098/rsos.220638},
	abstract = {Causal machine learning (CML) has experienced increasing popularity in healthcare. Beyond the inherent capabilities of adding domain knowledge into learning systems, CML provides a complete toolset for investigating how a system would react to an intervention (e.g. outcome given a treatment). Quantifying effects of interventions allows actionable decisions to be made while maintaining robustness in the presence of confounders. Here, we explore how causal inference can be incorporated into different aspects of clinical decision support systems by using recent advances in machine learning. Throughout this paper, we use Alzheimer’s disease to create examples for illustrating how CML can be advantageous in clinical scenarios. Furthermore, we discuss important challenges present in healthcare applications such as processing high-dimensional and unstructured data, generalization to out-of-distribution samples and temporal relationships, that despite the great effort from the research community remain to be solved. Finally, we review lines of research within causal representation learning, causal discovery and causal reasoning which offer the potential towards addressing the aforementioned challenges.},
	language = {en},
	number = {8},
	urldate = {2023-02-28},
	journal = {Royal Society Open Science},
	author = {Sanchez, Pedro and Voisey, Jeremy P. and Xia, Tian and Watson, Hannah I. and O’Neil, Alison Q. and Tsaftaris, Sotirios A.},
	month = aug,
	year = {2022},
	pages = {220638},
	file = {Sanchez et al. - 2022 - Causal machine learning for healthcare and precisi.pdf:/Users/jacquelinemaasch/Zotero/storage/A8MZAV6R/Sanchez et al. - 2022 - Causal machine learning for healthcare and precisi.pdf:application/pdf},
}

@article{bach_doubleml_nodate,
	title = {{DoubleML} – {An} {Object}-{Oriented} {Implementation} of {Double} {Machine} {Learning} in {Python}},
	abstract = {DoubleML is an open-source Python library implementing the double machine learning framework of Chernozhukov et al. (2018) for a variety of causal models. It contains functionalities for valid statistical inference on causal parameters when the estimation of nuisance parameters is based on machine learning methods. The object-oriented implementation of DoubleML provides a high ﬂexibility in terms of model speciﬁcations and makes it easily extendable. The package is distributed under the MIT license and relies on core libraries from the scientiﬁc Python ecosystem: scikit-learn, numpy, pandas, scipy, statsmodels and joblib. Source code, documentation and an extensive user guide can be found at https://github.com/DoubleML/doubleml-for-py and https://docs.doubleml.org.},
	language = {en},
	author = {Bach, Philipp and Chernozhukov, Victor and Kurz, Malte S and Spindler, Martin},
	file = {Bach et al. - DoubleML – An Object-Oriented Implementation of Do.pdf:/Users/jacquelinemaasch/Zotero/storage/W5RXC4HA/Bach et al. - DoubleML – An Object-Oriented Implementation of Do.pdf:application/pdf},
}

@misc{chernozhukov_doubledebiased_2017,
	title = {Double/{Debiased} {Machine} {Learning} for {Treatment} and {Causal} {Parameters}},
	url = {http://arxiv.org/abs/1608.00060},
	abstract = {We revisit the classic semiparametric problem of inference on a low dimensional parameter θ0 in the presence of high-dimensional nuisance parameters η0. We depart from the classical setting by allowing for η0 to be so high-dimensional that the traditional assumptions, such as Donsker properties, that limit complexity of the parameter space for this object break down. To estimate η0, we consider the use of statistical or machine learning (ML) methods which are particularly well-suited to estimation in modern, very high-dimensional cases. ML methods perform well by employing regularization to reduce variance and trading oﬀ regularization bias with overﬁtting in practice. However, both regularization bias and overﬁtting in estimating η0 cause a heavy bias in estimators of θ0 that are obtained by naively plugging ML estimators of η0 into estimating equations for θ0. This bias results in the naive estimator failing to be N −1/2 consistent, where N is the sample size. We show that the impact of regularization bias and overﬁtting on estimation of the parameter of interest θ0 can be removed by using two simple, yet critical, ingredients: (1) using Neyman-orthogonal moments/scores that have reduced sensitivity with respect to nuisance parameters to estimate θ0, and (2) making use of cross-ﬁtting which provides an eﬃcient form of data-splitting. We call the resulting set of methods double or debiased ML (DML). We verify that DML delivers point estimators that concentrate in a N −1/2-neighborhood of the true parameter values and are approximately unbiased and normally distributed, which allows construction of valid conﬁdence statements. The generic statistical theory of DML is elementary and simultaneously relies on only weak theoretical requirements which will admit the use of a broad array of modern ML methods for estimating the nuisance parameters such as random forests, lasso, ridge, deep neural nets, boosted trees, and various hybrids and ensembles of these methods. We illustrate the general theory by applying it to provide theoretical properties of DML applied to learn the main regression parameter in a partially linear regression model, DML applied to learn the coeﬃcient on an endogenous variable in a partially linear instrumental variables model, DML applied to learn the average treatment eﬀect and the average treatment eﬀect on the treated under unconfoundedness, and DML applied to learn the local average treatment eﬀect in an instrumental variables setting. In addition to these theoretical applications, we also illustrate the use of DML in three empirical examples.},
	language = {en},
	urldate = {2023-03-01},
	publisher = {arXiv},
	author = {Chernozhukov, Victor and Chetverikov, Denis and Demirer, Mert and Duflo, Esther and Hansen, Christian and Newey, Whitney and Robins, James},
	month = dec,
	year = {2017},
	note = {arXiv:1608.00060 [econ, stat]},
	keywords = {Statistics - Machine Learning, Economics - Econometrics, 62G},
	file = {Chernozhukov et al. - 2017 - DoubleDebiased Machine Learning for Treatment and.pdf:/Users/jacquelinemaasch/Zotero/storage/XYDYIRLD/Chernozhukov et al. - 2017 - DoubleDebiased Machine Learning for Treatment and.pdf:application/pdf},
}

@inproceedings{kasirzadeh_use_2021,
	address = {Virtual Event Canada},
	title = {The {Use} and {Misuse} of {Counterfactuals} in {Ethical} {Machine} {Learning}},
	isbn = {978-1-4503-8309-7},
	url = {https://dl.acm.org/doi/10.1145/3442188.3445886},
	doi = {10.1145/3442188.3445886},
	abstract = {The use of counterfactuals for considerations of algorithmic fairness and explainability is gaining prominence within the machine learning community and industry. This paper argues for more caution with the use of counterfactuals when the facts to be considered are social categories such as race or gender. We review a broad body of papers from philosophy and social sciences on social ontology and the semantics of counterfactuals, and we conclude that the counterfactual approach in machine learning fairness and social explainability can require an incoherent theory of what social categories are. Our findings suggest that most often the social categories may not admit counterfactual manipulation, and hence may not appropriately satisfy the demands for evaluating the truth or falsity of counterfactuals. This is important because the widespread use of counterfactuals in machine learning can lead to misleading results when applied in high-stakes domains. Accordingly, we argue that even though counterfactuals play an essential part in some causal inferences, their use for questions of algorithmic fairness and social explanations can create more problems than they resolve. Our positive result is a set of tenets about using counterfactuals for fairness and explanations in machine learning.},
	language = {en},
	urldate = {2023-03-01},
	booktitle = {Proceedings of the 2021 {ACM} {Conference} on {Fairness}, {Accountability}, and {Transparency}},
	publisher = {ACM},
	author = {Kasirzadeh, Atoosa and Smart, Andrew},
	month = mar,
	year = {2021},
	pages = {228--236},
	file = {Kasirzadeh and Smart - 2021 - The Use and Misuse of Counterfactuals in Ethical M.pdf:/Users/jacquelinemaasch/Zotero/storage/6XFCQESC/Kasirzadeh and Smart - 2021 - The Use and Misuse of Counterfactuals in Ethical M.pdf:application/pdf},
}
@article{park2020identifiability,
  title={Identifiability of additive noise models using conditional variances},
  author={Park, Gunwoong},
  journal={Journal of Machine Learning Research},
  volume={21},
  number={75},
  pages={1--34},
  year={2020}
}

@article{montagna2025score,
  title={Score matching through the roof: linear, nonlinear, and latent variables causal discovery},
  author={Montagna, Francesco and Faller, Philipp M and Bloebaum, Patrick and Kirschbaum, Elke and Locatello, Francesco},
  journal={Causal Learning and Reasoning (CLeaR), PMLR.},
  year={2025}
}

@article{jung_estimating_2021,
	title = {Estimating {Identifiable} {Causal} {Effects} through {Double} {Machine} {Learning}},
	volume = {35},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/17438},
	doi = {10.1609/aaai.v35i13.17438},
	abstract = {Identifying causal effects from observational data is a pervasive challenge found throughout the empirical sciences. Very general methods have been developed to decide the identiﬁability of a causal quantity from a combination of observational data and causal knowledge about the underlying system. In practice, however, there are still challenges to estimating identiﬁable causal functionals from ﬁnite samples. Recently, a method known as double/debiased machine learning (DML) (Chernozhukov et al. 2018) has been proposed to learn parameters leveraging modern machine learning techniques, which is both robust to model misspeciﬁcation and bias-reducing. Still, DML has only been used for causal estimation in settings when the back-door condition (also known as conditional ignorability) holds. In this paper, we develop a new, general class of estimators for any identiﬁable causal functionals that exhibit DML properties, which we name DML-ID. In particular, we introduce a complete identiﬁcation algorithm that returns an inﬂuence function (IF) for any identiﬁable causal functional. We then construct the DML estimator based on the derived IF. We show that DML-ID estimators hold the key properties of debiasedness and doubly robustness. Simulation results corroborate with the theory.},
	language = {en},
	number = {13},
	urldate = {2023-03-01},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Jung, Yonghan and Tian, Jin and Bareinboim, Elias},
	month = may,
	year = {2021},
	pages = {12113--12122},
	file = {Jung et al. - 2021 - Estimating Identifiable Causal Effects through Dou.pdf:/Users/jacquelinemaasch/Zotero/storage/NFAIHE2E/Jung et al. - 2021 - Estimating Identifiable Causal Effects through Dou.pdf:application/pdf},
}

@article{dickerman_counterfactual_2020,
	title = {Counterfactual prediction is not only for causal inference},
	volume = {35},
	issn = {0393-2990, 1573-7284},
	url = {https://link.springer.com/10.1007/s10654-020-00659-8},
	doi = {10.1007/s10654-020-00659-8},
	language = {en},
	number = {7},
	urldate = {2023-03-01},
	journal = {European Journal of Epidemiology},
	author = {Dickerman, Barbra A. and Hernán, Miguel A.},
	month = jul,
	year = {2020},
	pages = {615--617},
	file = {Dickerman and Hernán - 2020 - Counterfactual prediction is not only for causal i.pdf:/Users/jacquelinemaasch/Zotero/storage/FYXNSKLH/Dickerman and Hernán - 2020 - Counterfactual prediction is not only for causal i.pdf:application/pdf},
}

@article{verma_counterfactual_nodate,
	title = {Counterfactual {Explanations} and {Algorithmic} {Recourses} for {Machine} {Learning}: {A} {Review}},
	language = {en},
	author = {Verma, Sahil and Boonsanong, Varich and Hoang, Minh and Hines, Keegan E and Dickerson, John P and Shah, Chirag},
	file = {Verma et al. - Counterfactual Explanations and Algorithmic Recour.pdf:/Users/jacquelinemaasch/Zotero/storage/IL6XNW9R/Verma et al. - Counterfactual Explanations and Algorithmic Recour.pdf:application/pdf},
}

@book{neal_introduction_2020,
	title = {Introduction to {Causal} {Inference}},
	url = {https://www.bradyneal.com/Introduction_to_Causal_Inference-Aug27_2020-Neal.pdf},
	language = {en},
	author = {Neal, Brady},
	year = {2020},
	file = {Neal - Introduction to Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/DH3UF8VP/Neal - Introduction to Causal Inference.pdf:application/pdf},
}

@article{janzing_quantifying_2013,
	title = {Quantifying causal influences},
	volume = {41},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-41/issue-5/Quantifying-causal-influences/10.1214/13-AOS1145.full},
	doi = {10.1214/13-AOS1145},
	language = {en},
	number = {5},
	urldate = {2023-03-02},
	journal = {The Annals of Statistics},
	author = {Janzing, Dominik and Balduzzi, David and Grosse-Wentrup, Moritz and Schölkopf, Bernhard},
	month = oct,
	year = {2013},
	file = {Janzing et al. - 2013 - Quantifying causal influences.pdf:/Users/jacquelinemaasch/Zotero/storage/IST8ASXB/Janzing et al. - 2013 - Quantifying causal influences.pdf:application/pdf},
}

@article{pawlowski_deep_nodate,
	title = {Deep {Structural} {Causal} {Models} for {Tractable} {Counterfactual} {Inference}},
	abstract = {We formulate a general framework for building structural causal models (SCMs) with deep learning components. The proposed approach employs normalising ﬂows and variational inference to enable tractable inference of exogenous noise variables—a crucial step for counterfactual inference that is missing from existing deep causal learning methods. Our framework is validated on a synthetic dataset built on MNIST as well as on a real-world medical dataset of brain MRI scans. Our experimental results indicate that we can successfully train deep SCMs that are capable of all three levels of Pearl’s ladder of causation: association, intervention, and counterfactuals, giving rise to a powerful new approach for answering causal questions in imaging applications and beyond. The code for all our experiments is available at https://github.com/biomedia-mira/deepscm.},
	language = {en},
	author = {Pawlowski, Nick and Castro, Daniel C and Glocker, Ben},
	file = {Pawlowski et al. - Deep Structural Causal Models for Tractable Counte.pdf:/Users/jacquelinemaasch/Zotero/storage/LDCBRN8K/Pawlowski et al. - Deep Structural Causal Models for Tractable Counte.pdf:application/pdf},
}

@techreport{zang_high-throughput_2022,
	type = {preprint},
	title = {High-{Throughput} {Clinical} {Trial} {Emulation} with {Real} {World} {Data} and {Machine} {Learning}: {A} {Case} {Study} of {Drug} {Repurposing} for {Alzheimer}’s {Disease}},
	shorttitle = {High-{Throughput} {Clinical} {Trial} {Emulation} with {Real} {World} {Data} and {Machine} {Learning}},
	url = {http://medrxiv.org/lookup/doi/10.1101/2022.01.31.22270132},
	abstract = {ABSTRACT
          Clinical trial emulation, which is the process of mimicking targeted randomized controlled trials (RCT) with real-world data (RWD), has attracted growing attention and interest in recent years from the pharmaceutical industry. Different from RCTs which have stringent eligibility criteria for recruiting participants, RWD are more representative of real-world patients to whom the drugs will be prescribed. One technical challenge for trial emulation is how to conduct effective confounding control with complex RWD so that the treatment effects can be objectively derived. Recently many approaches, including deep learning algorithms, have been proposed for this goal, but there is still no systematic evaluation and practical guidance on them. In this paper, we emulate 430, 000 trials from two large-scale RWD warehouses, covering both electronic health records (EHR) and general claims, over 170 million patients spanning more than 10 years, aiming to identify new indications of approved drugs for Alzheimer’s disease (AD). We have investigated the behaviors of multiple different approaches including logistic regression and deep learning models, and propose a new model selection strategy that can significantly improve the performance of confounding balance of the participants in different arms of emulated trials. We demonstrate that regularized logistic regression-based propensity score (PS) model outperforms the deep learning-based PS model and others, which contradicts with our intuitions to a certain extent. Finally, we identified 8 drugs whose original indications are not AD (pantoprazole, gabapentin, acetaminophen, atorvastatin, albuterol, fluticasone, amoxicillin, and omeprazole), hold great potential of being beneficial to AD patients.},
	language = {en},
	urldate = {2023-03-02},
	institution = {Health Informatics},
	author = {Zang, Chengxi and Zhang, Hao and Xu, Jie and Zhang, Hansi and Fouladvand, Sajjad and Havaldar, Shreyas and Cheng, Feixiong and Chen, Kun and Chen, Yong and Glicksberg, Benjamin S. and Chen, Jin and Bian, Jiang and Wang, Fei},
	month = feb,
	year = {2022},
	doi = {10.1101/2022.01.31.22270132},
	file = {Zang et al. - 2022 - High-Throughput Clinical Trial Emulation with Real.pdf:/Users/jacquelinemaasch/Zotero/storage/FDASPII3/Zang et al. - 2022 - High-Throughput Clinical Trial Emulation with Real.pdf:application/pdf},
}

@article{tjoa_survey_2021,
	title = {A {Survey} on {Explainable} {Artificial} {Intelligence} ({XAI}): {Towards} {Medical} {XAI}},
	volume = {32},
	issn = {2162-237X, 2162-2388},
	shorttitle = {A {Survey} on {Explainable} {Artificial} {Intelligence} ({XAI})},
	url = {http://arxiv.org/abs/1907.07374},
	doi = {10.1109/TNNLS.2020.3027314},
	abstract = {Recently, artiﬁcial intelligence and machine learning in general have demonstrated remarkable performances in many tasks, from image processing to natural language processing, especially with the advent of deep learning. Along with research progress, they have encroached upon many different ﬁelds and disciplines. Some of them require high level of accountability and thus transparency, for example the medical sector. Explanations for machine decisions and predictions are thus needed to justify their reliability. This requires greater interpretability, which often means we need to understand the mechanism underlying the algorithms. Unfortunately, the blackbox nature of the deep learning is still unresolved, and many machine decisions are still poorly understood. We provide a review on interpretabilities suggested by different research works and categorize them. The different categories show different dimensions in interpretability research, from approaches that provide “obviously” interpretable information to the studies of complex patterns. By applying the same categorization to interpretability in medical research, it is hoped that (1) clinicians and practitioners can subsequently approach these methods with caution, (2) insights into interpretability will be born with more considerations for medical practices, and (3) initiatives to push forward data-based, mathematically- and technically-grounded medical education are encouraged.},
	language = {en},
	number = {11},
	urldate = {2023-03-02},
	journal = {IEEE Transactions on Neural Networks and Learning Systems},
	author = {Tjoa, Erico and Guan, Cuntai},
	month = nov,
	year = {2021},
	note = {arXiv:1907.07374 [cs]},
	keywords = {Computer Science - Machine Learning, Computer Science - Artificial Intelligence},
	pages = {4793--4813},
	file = {Tjoa and Guan - 2021 - A Survey on Explainable Artificial Intelligence (X.pdf:/Users/jacquelinemaasch/Zotero/storage/7KC7QPQK/Tjoa and Guan - 2021 - A Survey on Explainable Artificial Intelligence (X.pdf:application/pdf},
}

@inproceedings{abrate_counterfactual_2021,
	address = {Virtual Event Singapore},
	title = {Counterfactual {Graphs} for {Explainable} {Classification} of {Brain} {Networks}},
	isbn = {978-1-4503-8332-5},
	url = {https://dl.acm.org/doi/10.1145/3447548.3467154},
	doi = {10.1145/3447548.3467154},
	abstract = {Training graph classifiers able to distinguish between healthy brains and dysfunctional ones, can help identifying substructures associated to specific cognitive phenotypes. However, the mere predictive power of the graph classifier is of limited interest to the neuroscientists, which have plenty of tools for the diagnosis of specific mental disorders. What matters is the interpretation of the model, as it can provide novel insights and new hypotheses.},
	language = {en},
	urldate = {2023-03-02},
	booktitle = {Proceedings of the 27th {ACM} {SIGKDD} {Conference} on {Knowledge} {Discovery} \& {Data} {Mining}},
	publisher = {ACM},
	author = {Abrate, Carlo and Bonchi, Francesco},
	month = aug,
	year = {2021},
	pages = {2495--2504},
	file = {Abrate and Bonchi - 2021 - Counterfactual Graphs for Explainable Classificati.pdf:/Users/jacquelinemaasch/Zotero/storage/8EEEVLBM/Abrate and Bonchi - 2021 - Counterfactual Graphs for Explainable Classificati.pdf:application/pdf},
}

@article{marchezini_counterfactual_2022,
	title = {Counterfactual inference with latent variable and its application in mental health care},
	volume = {36},
	issn = {1384-5810, 1573-756X},
	url = {https://link.springer.com/10.1007/s10618-021-00818-9},
	doi = {10.1007/s10618-021-00818-9},
	language = {en},
	number = {2},
	urldate = {2023-03-02},
	journal = {Data Mining and Knowledge Discovery},
	author = {Marchezini, Guilherme F. and Lacerda, Anisio M. and Pappa, Gisele L. and Meira, Wagner and Miranda, Debora and Romano-Silva, Marco A. and Costa, Danielle S. and Diniz, Leandro Malloy},
	month = mar,
	year = {2022},
	pages = {811--840},
	file = {Marchezini et al. - 2022 - Counterfactual inference with latent variable and .pdf:/Users/jacquelinemaasch/Zotero/storage/VASVUPI2/Marchezini et al. - 2022 - Counterfactual inference with latent variable and .pdf:application/pdf},
}

@article{matthews_target_2022,
	title = {Target trial emulation: applying principles of randomised trials to observational studies},
	issn = {1756-1833},
	shorttitle = {Target trial emulation},
	url = {https://www.bmj.com/lookup/doi/10.1136/bmj-2022-071108},
	doi = {10.1136/bmj-2022-071108},
	language = {en},
	urldate = {2023-03-02},
	journal = {BMJ},
	author = {Matthews, Anthony A and Danaei, Goodarz and Islam, Nazrul and Kurth, Tobias},
	month = aug,
	year = {2022},
	pages = {e071108},
	file = {Matthews et al. - 2022 - Target trial emulation applying principles of ran.pdf:/Users/jacquelinemaasch/Zotero/storage/5PLBH6HT/Matthews et al. - 2022 - Target trial emulation applying principles of ran.pdf:application/pdf},
}

@article{oganisian_practical_2021,
	title = {A practical introduction to {Bayesian} estimation of causal effects: {Parametric} and nonparametric approaches},
	volume = {40},
	issn = {0277-6715, 1097-0258},
	shorttitle = {A practical introduction to {Bayesian} estimation of causal effects},
	url = {https://onlinelibrary.wiley.com/doi/10.1002/sim.8761},
	doi = {10.1002/sim.8761},
	abstract = {Substantial advances in Bayesian methods for causal inference have been made in recent years. We provide an introduction to Bayesian inference for causal effects for practicing statisticians who have some familiarity with Bayesian models and would like an overview of what it can add to causal estimation in practical settings. In the paper, we demonstrate how priors can induce shrinkage and sparsity in parametric models and be used to perform probabilistic sensitivity analyses around causal assumptions. We provide an overview of nonparametric Bayesian estimation and survey their applications in the causal inference literature. Inference in the point-treatment and time-varying treatment settings are considered. For the latter, we explore both static and dynamic treatment regimes. Throughout, we illustrate implementation using off-the-shelf open source software. We hope to leave the reader with implementation-level knowledge of Bayesian causal inference using both parametric and nonparametric models. All synthetic examples and code used in the paper are publicly available on a companion GitHub repository.},
	language = {en},
	number = {2},
	urldate = {2023-03-02},
	journal = {Statistics in Medicine},
	author = {Oganisian, Arman and Roy, Jason A.},
	month = jan,
	year = {2021},
	pages = {518--551},
	file = {Oganisian and Roy - 2021 - A practical introduction to Bayesian estimation of.pdf:/Users/jacquelinemaasch/Zotero/storage/DR5BQ4BR/Oganisian and Roy - 2021 - A practical introduction to Bayesian estimation of.pdf:application/pdf},
}

@article{scribe_3_nodate,
	title = {3: {Directed} {Graphical} {Models} ({Bayesian} {Networks})},
	language = {en},
	journal = {Bayesian Networks},
	author = {Scribe, Eric P Xing and Pokharel, Suman and Yang, Wendy and Yan, Donghui and Tang, Jingjing and Sun, Wenhuan},
	file = {Scribe et al. - 3 Directed Graphical Models (Bayesian Networks).pdf:/Users/jacquelinemaasch/Zotero/storage/LV5SKK7U/Scribe et al. - 3 Directed Graphical Models (Bayesian Networks).pdf:application/pdf},
}

@article{zhang_causal_nodate,
	title = {Causal {Discovery} with {Linear} {Non}-{Gaussian} {Models} under {Measurement} {Error}: {Structural} {Identiﬁability} {Results}},
	abstract = {Causal discovery methods aim to recover the causal process that generated purely observational data. Despite its successes on a number of real problems, the presence of measurement error in the observed data can produce serious mistakes in the output of various causal discovery methods. Given the ubiquity of measurement error caused by instruments or proxies used in the measuring process, this problem is one of the main obstacles to reliable causal discovery. It is still unknown to what extent the causal structure of relevant variables can be identiﬁed in principle. This study aims to take a step towards ﬁlling that void. We assume that the underlining process or the measurement-error free variables follows a linear, non-Guassian causal model, and show that the so-called ordered group decomposition of the causal model, which contains major causal information, is identiﬁable. The causal structure identiﬁability is further improved with diﬀerent types of sparsity constraints on the causal structure. Finally, we give rather mild conditions under which the whole causal structure is fully identiﬁable.},
	language = {en},
	author = {Zhang, Kun and Gong, Mingming and Ramsey, Joseph and Batmanghelich, Kayhan and Spirtes, Peter and Glymour, Clark},
	file = {Zhang et al. - Causal Discovery with Linear Non-Gaussian Models u.pdf:/Users/jacquelinemaasch/Zotero/storage/82ACUV3H/Zhang et al. - Causal Discovery with Linear Non-Gaussian Models u.pdf:application/pdf},
}

@misc{lachapelle_gradient-based_2020,
	title = {Gradient-{Based} {Neural} {DAG} {Learning}},
	url = {http://arxiv.org/abs/1906.02226},
	abstract = {We propose a novel score-based approach to learning a directed acyclic graph (DAG) from observational data. We adapt a recently proposed continuous constrained optimization formulation to allow for nonlinear relationships between variables using neural networks. This extension allows to model complex interactions while avoiding the combinatorial nature of the problem. In addition to comparing our method to existing continuous optimization methods, we provide missing empirical comparisons to nonlinear greedy search methods. On both synthetic and real-world data sets, this new method outperforms current continuous methods on most tasks, while being competitive with existing greedy search methods on important metrics for causal inference.},
	language = {en},
	urldate = {2023-03-02},
	publisher = {arXiv},
	author = {Lachapelle, Sébastien and Brouillard, Philippe and Deleu, Tristan and Lacoste-Julien, Simon},
	month = feb,
	year = {2020},
	note = {arXiv:1906.02226 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, I.2.6, I.5.1},
	file = {Lachapelle et al. - 2020 - Gradient-Based Neural DAG Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/62E5H3MD/Lachapelle et al. - 2020 - Gradient-Based Neural DAG Learning.pdf:application/pdf},
}

@inproceedings{solo_causality_2008,
	address = {Cancun, Mexico},
	title = {On causality and mutual information},
	isbn = {978-1-4244-3123-6},
	url = {http://ieeexplore.ieee.org/document/4738640/},
	doi = {10.1109/CDC.2008.4738640},
	abstract = {We provide for the ﬁrst time a formulation of nonlinear, nonstationary causality in terms of mutual information. We provide two fundamental mutual information identities; the ﬁrst relating Granger type causality to Sims type causality. The second providing a decomposition of mutual information into a sum of Granger and Sims type terms. We also develop asymptotic relations that emerge under strict stationarity and generalise earlier work of Geweke. We relate our work in general with earlier developments.},
	language = {en},
	urldate = {2023-03-03},
	booktitle = {2008 47th {IEEE} {Conference} on {Decision} and {Control}},
	publisher = {IEEE},
	author = {Solo, Victor},
	year = {2008},
	pages = {4939--4944},
	file = {Solo - 2008 - On causality and mutual information.pdf:/Users/jacquelinemaasch/Zotero/storage/X4UHSB5S/Solo - 2008 - On causality and mutual information.pdf:application/pdf},
}

@article{yamada_dependence_2010,
	title = {Dependence {Minimizing} {Regression} with {Model} {Selection} for {Non}-{Linear} {Causal} {Inference} under {Non}-{Gaussian} {Noise}},
	volume = {24},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/7655},
	doi = {10.1609/aaai.v24i1.7655},
	abstract = {The discovery of non-linear causal relationship under additive non-Gaussian noise models has attracted considerable attention recently because of their high ﬂexibility. In this paper, we propose a novel causal inference algorithm called least-squares independence regression (LSIR). LSIR learns the additive noise model through minimization of an estimator of the squaredloss mutual information between inputs and residuals. A notable advantage of LSIR over existing approaches is that tuning parameters such as the kernel width and the regularization parameter can be naturally optimized by cross-validation, allowing us to avoid overﬁtting in a data-dependent fashion. Through experiments with real-world datasets, we show that LSIR compares favorably with the state-of-the-art causal inference method.},
	language = {en},
	number = {1},
	urldate = {2023-03-03},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Yamada, Makoto and Sugiyama, Masashi},
	month = jul,
	year = {2010},
	pages = {643--648},
	file = {Yamada and Sugiyama - 2010 - Dependence Minimizing Regression with Model Select.pdf:/Users/jacquelinemaasch/Zotero/storage/U5XPZ78N/Yamada and Sugiyama - 2010 - Dependence Minimizing Regression with Model Select.pdf:application/pdf},
}

@misc{ranganath_multiple_2019,
	title = {Multiple {Causal} {Inference} with {Latent} {Confounding}},
	url = {http://arxiv.org/abs/1805.08273},
	abstract = {Causal inference from observational data requires assumptions. These assumptions range from measuring confounders to identifying instruments. Traditionally, causal inference assumptions have focused on estimation of effects for a single treatment. In this work, we construct techniques for estimation with multiple treatments in the presence of unobserved confounding. We develop two assumptions based on shared confounding between treatments and independence of treatments given the confounder. Together, these assumptions lead to a confounder estimator regularized by mutual information. For this estimator, we develop a tractable lower bound. To recover treatment effects, we use the residual information in the treatments independent of the confounder. We validate on simulations and an example from clinical medicine.},
	language = {en},
	urldate = {2023-03-03},
	publisher = {arXiv},
	author = {Ranganath, Rajesh and Perotte, Adler},
	month = mar,
	year = {2019},
	note = {arXiv:1805.08273 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Ranganath and Perotte - 2019 - Multiple Causal Inference with Latent Confounding.pdf:/Users/jacquelinemaasch/Zotero/storage/DWRN8V4F/Ranganath and Perotte - 2019 - Multiple Causal Inference with Latent Confounding.pdf:application/pdf},
}

@article{zhang_estimation_2016,
	title = {On {Estimation} of {Functional} {Causal} {Models}: {General} {Results} and {Application} to the {Post}-{Nonlinear} {Causal} {Model}},
	volume = {7},
	issn = {2157-6904, 2157-6912},
	shorttitle = {On {Estimation} of {Functional} {Causal} {Models}},
	url = {https://dl.acm.org/doi/10.1145/2700476},
	doi = {10.1145/2700476},
	abstract = {Compared to constraint-based causal discovery, causal discovery based on functional causal models is able to identify the whole causal model under appropriate assumptions [Shimizu et al. 2006; Hoyer et al. 2009; Zhang and Hyvärinen 2009b]. Functional causal models represent the effect as a function of the direct causes together with an independent noise term. Examples include the linear non-Gaussian acyclic model (LiNGAM), nonlinear additive noise model, and post-nonlinear (PNL) model. Currently, there are two ways to estimate the parameters in the models: dependence minimization and maximum likelihood. In this article, we show that for any acyclic functional causal model, minimizing the mutual information between the hypothetical cause and the noise term is equivalent to maximizing the data likelihood with a flexible model for the distribution of the noise term. We then focus on estimation of the PNL causal model and propose to estimate it with the warped Gaussian process with the noise modeled by the mixture of Gaussians. As a Bayesian nonparametric approach, it outperforms the previous one based on mutual information minimization with nonlinear functions represented by multilayer perceptrons; we also show that unlike the ordinary regression, estimation results of the PNL causal model are sensitive to the assumption on the noise distribution. Experimental results on both synthetic and real data support our theoretical claims.},
	language = {en},
	number = {2},
	urldate = {2023-03-03},
	journal = {ACM Transactions on Intelligent Systems and Technology},
	author = {Zhang, Kun and Wang, Zhikun and Zhang, Jiji and Schölkopf, Bernhard},
	month = jan,
	year = {2016},
	pages = {1--22},
	file = {Zhang et al. - 2016 - On Estimation of Functional Causal Models General.pdf:/Users/jacquelinemaasch/Zotero/storage/MLMD8AK2/Zhang et al. - 2016 - On Estimation of Functional Causal Models General.pdf:application/pdf},
}

@article{sugiyama_machine_2012,
	title = {Machine {Learning} with {Squared}-{Loss} {Mutual} {Information}},
	volume = {15},
	issn = {1099-4300},
	url = {http://www.mdpi.com/1099-4300/15/1/80},
	doi = {10.3390/e15010080},
	abstract = {Mutual information (MI) is useful for detecting statistical independence between random variables, and it has been successfully applied to solving various machine learning problems. Recently, an alternative to MI called squared-loss MI (SMI) was introduced. While ordinary MI is the Kullback–Leibler divergence from the joint distribution to the product of the marginal distributions, SMI is its Pearson divergence variant. Because both the divergences belong to the f -divergence family, they share similar theoretical properties. However, a notable advantage of SMI is that it can be approximated from data in a computationally more efﬁcient and numerically more stable way than ordinary MI. In this article, we review recent development in SMI approximation based on direct density-ratio estimation and SMI-based machine learning techniques such as independence testing, dimensionality reduction, canonical dependency analysis, independent component analysis, object matching, clustering, and causal inference.},
	language = {en},
	number = {1},
	urldate = {2023-03-03},
	journal = {Entropy},
	author = {Sugiyama, Masashi},
	month = dec,
	year = {2012},
	pages = {80--112},
	file = {Sugiyama - 2012 - Machine Learning with Squared-Loss Mutual Informat.pdf:/Users/jacquelinemaasch/Zotero/storage/W6YQEWIC/Sugiyama - 2012 - Machine Learning with Squared-Loss Mutual Informat.pdf:application/pdf},
}

@article{verdu_empirical_2019,
	title = {Empirical {Estimation} of {Information} {Measures}: {A} {Literature} {Guide}},
	volume = {21},
	issn = {1099-4300},
	shorttitle = {Empirical {Estimation} of {Information} {Measures}},
	url = {https://www.mdpi.com/1099-4300/21/8/720},
	doi = {10.3390/e21080720},
	abstract = {We give a brief survey of the literature on the empirical estimation of entropy, differential entropy, relative entropy, mutual information and related information measures. While those quantities are of central importance in information theory, universal algorithms for their estimation are increasingly important in data science, machine learning, biology, neuroscience, economics, language, and other experimental sciences.},
	language = {en},
	number = {8},
	urldate = {2023-03-03},
	journal = {Entropy},
	author = {Verdú, Sergio},
	month = jul,
	year = {2019},
	pages = {720},
	file = {Verdú - 2019 - Empirical Estimation of Information Measures A Li.pdf:/Users/jacquelinemaasch/Zotero/storage/252X8QL9/Verdú - 2019 - Empirical Estimation of Information Measures A Li.pdf:application/pdf},
}

@inproceedings{beraha_feature_2019,
	address = {Budapest, Hungary},
	title = {Feature {Selection} via {Mutual} {Information}: {New} {Theoretical} {Insights}},
	isbn = {978-1-72811-985-4},
	shorttitle = {Feature {Selection} via {Mutual} {Information}},
	url = {https://ieeexplore.ieee.org/document/8852410/},
	doi = {10.1109/IJCNN.2019.8852410},
	abstract = {Mutual information has been successfully adopted in ﬁlter feature-selection methods to assess both the relevancy of a subset of features in predicting the target variable and the redundancy with respect to other variables. However, existing algorithms are mostly heuristic and do not offer any guarantee on the proposed solution. In this paper, we provide novel theoretical results showing that conditional mutual information naturally arises when bounding the ideal regression/classiﬁcation errors achieved by different subsets of features. Leveraging on these insights, we propose a novel stopping condition for backward and forward greedy methods which ensures that the ideal prediction error using the selected feature subset remains bounded by a user-speciﬁed threshold. We provide numerical simulations to support our theoretical claims and compare to common heuristic methods.},
	language = {en},
	urldate = {2023-03-03},
	booktitle = {2019 {International} {Joint} {Conference} on {Neural} {Networks} ({IJCNN})},
	publisher = {IEEE},
	author = {Beraha, Mario and Metelli, Alberto Maria and Papini, Matteo and Tirinzoni, Andrea and Restelli, Marcello},
	month = jul,
	year = {2019},
	pages = {1--9},
	file = {Beraha et al. - 2019 - Feature Selection via Mutual Information New Theo.pdf:/Users/jacquelinemaasch/Zotero/storage/HP8Q7MUL/Beraha et al. - 2019 - Feature Selection via Mutual Information New Theo.pdf:application/pdf},
}

@article{kraskov_estimating_2004,
	title = {Estimating mutual information},
	volume = {69},
	issn = {1539-3755, 1550-2376},
	doi = {10.1103/PhysRevE.69.066138},
	language = {en},
	number = {6},
	urldate = {2023-03-03},
	journal = {Physical Review E},
	author = {Kraskov, Alexander and Stögbauer, Harald and Grassberger, Peter},
	month = jun,
	year = {2004},
	pages = {066138},
	file = {Kraskov et al. - 2004 - Estimating mutual information.pdf:/Users/jacquelinemaasch/Zotero/storage/EVPSCGEJ/Kraskov et al. - 2004 - Estimating mutual information.pdf:application/pdf},
}


@article{cover_elements_2006,
	title = {{ELEMENTS} {OF} {INFORMATION} {THEORY}},
	language = {en},
	author = {Cover, Thomas M and Thomas, Joy A},
	year = {2006},
	file = {Cover and Thomas - ELEMENTS OF INFORMATION THEORY.pdf:/Users/jacquelinemaasch/Zotero/storage/V6LTWVVS/Cover and Thomas - ELEMENTS OF INFORMATION THEORY.pdf:application/pdf},
}

@inproceedings{gao_estimating_2017,
	title = {Estimating {Mutual} {Information} for {Discrete}-{Continuous} {Mixtures}},
	abstract = {Estimation of mutual information from observed samples is a basic primitive in machine learning, useful in several learning tasks including correlation mining, information bottleneck, Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a quantity well-deﬁned for general probability spaces, estimators have been developed only in the special case of discrete or continuous pairs of random variables. Most of these estimators operate using the 3H-principle, i.e., by calculating the three (differential) entropies of X, Y and the pair (X, Y ). However, in general mixture spaces, such individual entropies are not well deﬁned, even though mutual information is. In this paper, we develop a novel estimator for estimating mutual information in discrete-continuous mixtures. We prove the consistency of this estimator theoretically as well as demonstrate its excellent empirical performance. This problem is relevant in a wide-array of applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.},
	language = {en},
	booktitle = {Proceedings of the 31st {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Gao, Weihao and Kannan, Sreeram and Oh, Sewoong and Viswanath, Pramod},
	year = {2017},
	file = {Gao et al. - Estimating Mutual Information for Discrete-Continu.pdf:/Users/jacquelinemaasch/Zotero/storage/VGNADWZK/Gao et al. - Estimating Mutual Information for Discrete-Continu.pdf:application/pdf},
}

@article{ai_testing_2022,
	title = {Testing unconditional and conditional independence via mutual information},
	issn = {03044076},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0304407622001609},
	doi = {10.1016/j.jeconom.2022.07.011},
	abstract = {Testing independence has garnered increasing attention in the econometric and statistical literature. Many tests have been proposed, most of which are inconsistent against all departures from independence. Few of those tests, though consistent, suffer a significant loss of local power. This study proposes a mutual information test for testing independence. The proposed test is simple to implement and, with a slight loss of local power, is consistent against all departures from independence. The key driving factor is that we estimate the density ratio directly. This value is constant in a state of independence. This is in contrast with related studies that estimate the joint and marginal density functions to form the density ratio. A small-scale simulation study indicates that the proposed test outperforms the existing alternatives in various dependence structures.},
	language = {en},
	urldate = {2023-03-03},
	journal = {Journal of Econometrics},
	author = {Ai, Chunrong and Sun, Li-Hsien and Zhang, Zheng and Zhu, Liping},
	month = sep,
	year = {2022},
	pages = {S0304407622001609},
	file = {Ai et al. - 2022 - Testing unconditional and conditional independence.pdf:/Users/jacquelinemaasch/Zotero/storage/S7TIECR2/Ai et al. - 2022 - Testing unconditional and conditional independence.pdf:application/pdf},
}

@misc{song_understanding_2020,
	title = {Understanding the {Limitations} of {Variational} {Mutual} {Information} {Estimators}},
	url = {http://arxiv.org/abs/1910.06222},
	abstract = {Variational approaches based on neural networks are showing promise for estimating mutual information (MI) between high dimensional variables. However, they can be difﬁcult to use in practice due to poorly understood bias/variance tradeoffs. We theoretically show that, under some conditions, estimators such as MINE exhibit variance that could grow exponentially with the true amount of underlying MI. We also empirically demonstrate that existing estimators fail to satisfy basic self-consistency properties of MI, such as data processing and additivity under independence. Based on a uniﬁed perspective of variational approaches, we develop a new estimator that focuses on variance reduction. Empirical results on standard benchmark tasks demonstrate that our proposed estimator exhibits improved biasvariance trade-offs on standard benchmark tasks.},
	language = {en},
	urldate = {2023-03-03},
	publisher = {arXiv},
	author = {Song, Jiaming and Ermon, Stefano},
	month = mar,
	year = {2020},
	note = {arXiv:1910.06222 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Information Theory},
	file = {Song and Ermon - 2020 - Understanding the Limitations of Variational Mutua.pdf:/Users/jacquelinemaasch/Zotero/storage/VVGDNQ4F/Song and Ermon - 2020 - Understanding the Limitations of Variational Mutua.pdf:application/pdf},
}

@article{berrett_nonparametric_2019,
	title = {Nonparametric independence testing via mutual information},
	volume = {106},
	issn = {0006-3444, 1464-3510},
	url = {https://academic.oup.com/biomet/article/106/3/547/5511208},
	doi = {10.1093/biomet/asz024},
	abstract = {We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently developed efﬁcient entropy estimators derived from nearest neighbour distances. The proposed critical values may be obtained by simulation in the case where an approximation to one marginal is available or by permuting the data otherwise. This facilitates size guarantees, and we provide local power analyses, uniformly over classes of densities whose mutual information satisﬁes a lower bound. Our ideas may be extended to provide new goodness-of-ﬁt tests for normal linear models based on assessing the independence of our vector of covariates and an appropriately deﬁned notion of an error vector. The theory is supported by numerical studies on both simulated and real data.},
	language = {en},
	number = {3},
	urldate = {2023-03-03},
	journal = {Biometrika},
	author = {Berrett, T B and Samworth, R J},
	month = sep,
	year = {2019},
	pages = {547--566},
	file = {Berrett and Samworth - 2019 - Nonparametric independence testing via mutual info.pdf:/Users/jacquelinemaasch/Zotero/storage/XVXSTPGS/Berrett and Samworth - 2019 - Nonparametric independence testing via mutual info.pdf:application/pdf},
}

@inproceedings{mukherjee_ccmi_2020,
	title = {{CCMI} : {Classiﬁer} based {Conditional} {Mutual} {Information} {Estimation}},
	volume = {115},
	abstract = {Conditional Mutual Information (CMI) is a measure of conditional dependence between random variables X and Y, given another random variable Z. It can be used to quantify conditional dependence among variables in many data-driven inference problems such as graphical models, causal learning, feature selection and time-series analysis. While k-nearest neighbor (kNN) based estimators as well as kernel-based methods have been widely used for CMI estimation, they suffer severely from the curse of dimensionality. In this paper, we leverage advances in classiﬁers and generative models to design methods for CMI estimation. Speciﬁcally, we introduce an estimator for KL-Divergence based on the likelihood ratio by training a classiﬁer to distinguish the observed joint distribution from the product distribution. We then show how to construct several CMI estimators using this basic divergence estimator by drawing ideas from conditional generative models. We demonstrate that the estimates from our proposed approaches do not degrade in performance with increasing dimension and obtain signiﬁcant improvement over the widely used KSG estimator. Finally, as an application of accurate CMI estimation, we use our best estimator for conditional independence testing and achieve superior performance than the state-of-the-art tester on both simulated and real data-sets.},
	language = {en},
	booktitle = {Proceedings of {The} 35th {Uncertainty} in {Artificial} {Intelligence} {Conference}},
	publisher = {PMLR},
	author = {Mukherjee, Sudipto and Asnani, Himanshu and Kannan, Sreeram},
	year = {2020},
	pages = {1083--1093},
	file = {Mukherjee et al. - CCMI  Classiﬁer based Conditional Mutual Informat.pdf:/Users/jacquelinemaasch/Zotero/storage/75C5CY8C/Mukherjee et al. - CCMI  Classiﬁer based Conditional Mutual Informat.pdf:application/pdf},
}

@inproceedings{runge_conditional_2018,
	address = {Lanzarote, Spain},
	title = {Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information},
	booktitle = {Proceedings of the 21st {International} {Conference} on {Artificial} {Intelligence} and {Statistics} ({AISTATS})},
	author = {Runge, Jakob},
	year = {2018},
	file = {Runge - 2018 - Conditional independence testing based on a neares.pdf:/Users/jacquelinemaasch/Zotero/storage/5D57TB9Y/Runge - 2018 - Conditional independence testing based on a neares.pdf:application/pdf},
}

@article{ross_mutual_2014,
	title = {Mutual {Information} between {Discrete} and {Continuous} {Data} {Sets}},
	volume = {9},
	issn = {1932-6203},
	url = {https://dx.plos.org/10.1371/journal.pone.0087357},
	doi = {10.1371/journal.pone.0087357},
	abstract = {Mutual information (MI) is a powerful method for detecting relationships between data sets. There are accurate methods for estimating MI that avoid problems with ‘‘binning’’ when both data sets are discrete or when both data sets are continuous. We present an accurate, non-binning MI estimator for the case of one discrete data set and one continuous data set. This case applies when measuring, for example, the relationship between base sequence and gene expression level, or the effect of a cancer drug on patient survival time. We also show how our method can be adapted to calculate the Jensen–Shannon divergence of two or more data sets.},
	language = {en},
	number = {2},
	urldate = {2023-03-03},
	journal = {PLoS ONE},
	author = {Ross, Brian C.},
	editor = {Marinazzo, Daniele},
	month = feb,
	year = {2014},
	pages = {e87357},
	file = {Ross - 2014 - Mutual Information between Discrete and Continuous.pdf:/Users/jacquelinemaasch/Zotero/storage/FJYZTPEI/Ross - 2014 - Mutual Information between Discrete and Continuous.pdf:application/pdf},
}

@article{lundberg_what_nodate,
	title = {What {Is} {Your} {Estimand}? {Defining} the {Target} {Quantity} {Connects} {Statistical} {Evidence} to {Theory}},
	abstract = {We make only one point in this article. Every quantitative study must be able to answer the question: what is your estimand? The estimand is the target quantity—the purpose of the statistical analysis. Much attention is already placed on how to do estimation; a similar degree of care should be given to defining the thing we are estimating. We advocate that authors state the central quantity of each analysis—the theoretical estimand—in precise terms that exist outside of any statistical model. In our framework, researchers do three things: (1) set a theoretical estimand, clearly connecting this quantity to theory; (2) link to an empirical estimand, which is informative about the theoretical estimand under some identification assumptions; and (3) learn from data. Adding precise estimands to research practice expands the space of theoretical questions, clarifies how evidence can speak to those questions, and unlocks new tools for estimation. By grounding all three steps in a precise statement of the target quantity, our framework connects statistical evidence to theory.},
	language = {en},
	journal = {American Sociological Review},
	author = {Lundberg, Ian and Johnson, Rebecca and Stewart, Brandon M},
	file = {Lundberg et al. - What Is Your Estimand Defining the Target Quantit.pdf:/Users/jacquelinemaasch/Zotero/storage/6EEFSHLY/Lundberg et al. - What Is Your Estimand Defining the Target Quantit.pdf:application/pdf},
}

@incollection{mckay_illari_error_2011,
	title = {The error term and its interpretation in structural models in econometrics},
	isbn = {978-0-19-957413-1},
	url = {https://academic.oup.com/book/3313/chapter/144354581},
	abstract = {This chapter explores what the error term represents in structural models in econometrics and the assumptions about the error terms that are used for successful statistical and causal inference. The error term is of particular interest because it acts as a coverall term for parts of the system that are not fully known about and not explicitly modelled. The chapter attempts to bring some of the key assumptions imposed on the error term for di erent purposes (statistical and causal inference) and to ask to what extent the conditions imposed on the error term can be empirically tested in some way.},
	language = {en},
	urldate = {2023-03-07},
	booktitle = {Causality in the {Sciences}},
	publisher = {Oxford University Press},
	author = {Fennell, Damien},
	editor = {McKay Illari, Phyllis and Russo, Federica and Williamson, Jon},
	month = mar,
	year = {2011},
	doi = {10.1093/acprof:oso/9780199574131.003.0017},
	doi = {10.1093/acprof:oso/9780199574131.003.0017},
	pages = {361--378},
	file = {Fennell - 2011 - The error term and its interpretation in structura.pdf:/Users/jacquelinemaasch/Zotero/storage/TGTFYCAY/Fennell - 2011 - The error term and its interpretation in structura.pdf:application/pdf},
}

@article{janes_quantifying_2010,
	title = {On quantifying the magnitude of confounding},
	volume = {11},
	issn = {1465-4644, 1468-4357},
	url = {https://academic.oup.com/biostatistics/article-lookup/doi/10.1093/biostatistics/kxq007},
	doi = {10.1093/biostatistics/kxq007},
	abstract = {When estimating the association between an exposure and outcome, a simple approach to quantifying the amount of confounding by a factor, Z , is to compare estimates of the exposure–outcome association with and without adjustment for Z . This approach is widely believed to be problematic due to the nonlinearity of some exposure-effect measures. When the expected value of the outcome is modeled as a nonlinear function of the exposure, the adjusted and unadjusted exposure effects can differ even in the absence of confounding (Greenland, Robins, and Pearl, 1999); we call this the nonlinearity effect. In this paper, we propose a corrected measure of confounding that does not include the nonlinearity effect. The performances of the simple and corrected estimates of confounding are assessed in simulations and illustrated using a study of risk factors for low birth–weight infants. We conclude that the simple estimate of confounding is adequate or even preferred in settings where the nonlinearity effect is very small. In settings with a sizable nonlinearity effect, the corrected estimate of confounding has improved performance.},
	language = {en},
	number = {3},
	urldate = {2023-03-08},
	journal = {Biostatistics},
	author = {Janes, H. and Dominici, F. and Zeger, S.},
	month = jul,
	year = {2010},
	pages = {572--582},
	file = {Janes et al. - 2010 - On quantifying the magnitude of confounding.pdf:/Users/jacquelinemaasch/Zotero/storage/DRMWUU8S/Janes et al. - 2010 - On quantifying the magnitude of confounding.pdf:application/pdf},
}

@article{darrow_confounding_2011,
	title = {Confounding and {Bias} in the {Attributable} {Fraction}},
	volume = {22},
	issn = {1044-3983},
	url = {https://journals.lww.com/00001648-201101000-00009},
	doi = {10.1097/EDE.0b013e3181fce49b},
	abstract = {Inappropriate methods are frequently used to calculate the population attributable fraction (AF) for a given exposure of interest. This commonly occurs when authors use adjusted relative risks (RRs) reported in the literature (the "source" data), without access to the original data. In this analysis, we examine the relationship between the direction and magnitude of confounding in the source data and resulting bias in the attributable fraction when incorrect methods are used. We assess confounding by the confounding risk ratio, which is the ratio of the crude RR to the adjusted RR. We assess bias in the AF by the ratio of the incorrectly calculated AF to the correctly calculated AF. Using generated data, we examine the relationship between confounding and AF bias under various sce? narios of population prevalence of exposure and strength of the exposure-disease association. For confounding risk ratios greater than 1.0 (ie, crude RR {\textgreater}adjusted RR), the AF is underestimated; for confounding risk ratios less than 1.0 (ie, crude RR {\textless}adjusted RR), the AF is overestimated. Bias in the AF increases as the magnitude of the confounding increases, and is dependent on the prevalence of exposure in the total population, with bias greatest at the lowest prevalence of exposure. Bias in the AF is also higher when the exposure-disease association is weaker. Results of these analyses can assist interpretation of incorrectly calculated attributable frac? tion estimates commonly reported in the epidemiologic literature.},
	language = {en},
	number = {1},
	urldate = {2023-03-08},
	journal = {Epidemiology},
	author = {Darrow, Lyndsey A. and Steenland, N. Kyle},
	month = jan,
	year = {2011},
	pages = {53--58},
	file = {Darrow and Steenland - 2011 - Confounding and Bias in the Attributable Fraction.pdf:/Users/jacquelinemaasch/Zotero/storage/B6IDL7GF/Darrow and Steenland - 2011 - Confounding and Bias in the Attributable Fraction.pdf:application/pdf},
}

@article{schuster_noncollapsibility_2021,
	title = {Noncollapsibility and its role in quantifying confounding bias in logistic regression},
	volume = {21},
	issn = {1471-2288},
	url = {https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-021-01316-8},
	doi = {10.1186/s12874-021-01316-8},
	abstract = {Background:  Confounding bias is a common concern in epidemiological research. Its presence is often determined by comparing exposure effects between univariable- and multivariable regression models, using an arbitrary threshold of a 10\% difference to indicate confounding bias. However, many clinical researchers are not aware that the use of this change-in-estimate criterion may lead to wrong conclusions when applied to logistic regression coefficients. This is due to a statistical phenomenon called noncollapsibility, which manifests itself in logistic regression models. This paper aims to clarify the role of noncollapsibility in logistic regression and to provide guidance in determining the presence of confounding bias.
Methods:  A Monte Carlo simulation study was designed to uncover patterns of confounding bias and noncollapsibility effects in logistic regression. An empirical data example was used to illustrate the inability of the change-in-estimate criterion to distinguish confounding bias from noncollapsibility effects.
Results:  The simulation study showed that, depending on the sign and magnitude of the confounding bias and the noncollapsibility effect, the difference between the effect estimates from univariable- and multivariable regression models may underestimate or overestimate the magnitude of the confounding bias. Because of the noncollapsibility effect, multivariable regression analysis and inverse probability weighting provided different but valid estimates of the confounder-adjusted exposure effect. In our data example, confounding bias was underestimated by the change in estimate due to the presence of a noncollapsibility effect.
Conclusion:  In logistic regression, the difference between the univariable- and multivariable effect estimate might not only reflect confounding bias but also a noncollapsibility effect. Ideally, the set of confounders is determined at the study design phase and based on subject matter knowledge. To quantify confounding bias, one could compare the unadjusted exposure effect estimate and the estimate from an inverse probability weighted model.},
	language = {en},
	number = {1},
	urldate = {2023-03-08},
	journal = {BMC Medical Research Methodology},
	author = {Schuster, Noah A. and Twisk, Jos W. R. and ter Riet, Gerben and Heymans, Martijn W. and Rijnhart, Judith J. M.},
	month = dec,
	year = {2021},
	pages = {136},
	file = {Schuster et al. - 2021 - Noncollapsibility and its role in quantifying conf.pdf:/Users/jacquelinemaasch/Zotero/storage/YR5BEWFN/Schuster et al. - 2021 - Noncollapsibility and its role in quantifying conf.pdf:application/pdf},
}

@article{miettinen_confounding_1981,
	title = {{CONFOUNDING}: {ESSENCE} {AND} {DETECTION1}},
	volume = {114},
	issn = {1476-6256, 0002-9262},
	shorttitle = {{CONFOUNDING}},
	url = {https://academic.oup.com/aje/article-lookup/doi/10.1093/oxfordjournals.aje.a113225},
	doi = {10.1093/oxfordjournals.aje.a113225},
	language = {en},
	number = {4},
	urldate = {2023-03-08},
	journal = {American Journal of Epidemiology},
	author = {Miettinen, Olli S. and Cook, E. Francis},
	month = oct,
	year = {1981},
	pages = {593--603},
	file = {Miettinen and Cook - 1981 - CONFOUNDING ESSENCE AND DETECTION1.pdf:/Users/jacquelinemaasch/Zotero/storage/CZ3Q43CZ/Miettinen and Cook - 1981 - CONFOUNDING ESSENCE AND DETECTION1.pdf:application/pdf},
}

@article{talbot_change_2021,
	title = {The change in estimate method for selecting confounders: {A} simulation study},
	volume = {30},
	issn = {0962-2802, 1477-0334},
	shorttitle = {The change in estimate method for selecting confounders},
	url = {http://journals.sagepub.com/doi/10.1177/09622802211034219},
	doi = {10.1177/09622802211034219},
	abstract = {Background: The change in estimate is a popular approach for selecting confounders in epidemiology. It is recommended in epidemiologic textbooks and articles over significance test of coefficients, but concerns have been raised concerning its validity. Few simulation studies have been conducted to investigate its performance.
Methods: An extensive simulation study was realized to compare different implementations of the change in estimate method. The implementations were also compared when estimating the association of body mass index with diastolic blood pressure in the PROspective Qu ebec Study on Work and Health.
Results: All methods were susceptible to introduce important bias and to produce confidence intervals that included the true effect much less often than expected in at least some scenarios. Overall mixed results were obtained regarding the accuracy of estimators, as measured by the mean squared error. No implementation adequately differentiated confounders from non-confounders. In the real data analysis, none of the implementation decreased the estimated standard error.
Conclusion: Based on these results, it is questionable whether change in estimate methods are beneficial in general, considering their low ability to improve the precision of estimates without introducing bias and inability to yield valid confidence intervals or to identify true confounders.},
	language = {en},
	number = {9},
	urldate = {2023-03-08},
	journal = {Statistical Methods in Medical Research},
	author = {Talbot, Denis and Diop, Awa and Lavigne-Robichaud, Mathilde and Brisson, Chantal},
	month = oct,
	year = {2021},
	pages = {2032--2044},
	file = {Talbot et al. - 2021 - The change in estimate method for selecting confou.pdf:/Users/jacquelinemaasch/Zotero/storage/FBMATE3G/Talbot et al. - 2021 - The change in estimate method for selecting confou.pdf:application/pdf},
}

@article{sun_revisiting_2020,
	title = {Revisiting {Probability} {Distribution} {Assumptions} for {Information} {Theoretic} {Feature} {Selection}},
	volume = {34},
	issn = {2374-3468, 2159-5399},
	url = {https://ojs.aaai.org/index.php/AAAI/article/view/6050},
	doi = {10.1609/aaai.v34i04.6050},
	abstract = {Feature selection has been shown to be beneﬁcial for many data mining and machine learning tasks, especially for big data analytics. Mutual Information (MI) is a well-known information-theoretic approach used to evaluate the relevance of feature subsets and class labels. However, estimating highdimensional MI poses signiﬁcant challenges. Consequently, a great deal of research has focused on using low-order MI approximations or computing a lower bound on MI called Variational Information (VI). These methods often require certain assumptions made on the probability distributions of features such that these distributions are realistic yet tractable to compute. In this paper, we reveal two sets of distribution assumptions underlying many MI and VI based methods: Feature Independence Distribution and Geometric Mean Distribution. We systematically analyze their strengths and weaknesses and propose a logical extension called Arithmetic Mean Distribution, which leads to an unbiased and normalised estimation of probability densities. We conduct detailed empirical studies across a suite of 29 real-world classiﬁcation problems and illustrate improved prediction accuracy of our methods based on the identiﬁcation of more informative features, thus providing support for our theoretical ﬁndings.},
	language = {en},
	number = {04},
	urldate = {2023-03-08},
	journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
	author = {Sun, Yuan and Wang, Wei and Kirley, Michael and Li, Xiaodong and Chan, Jeffrey},
	month = apr,
	year = {2020},
	pages = {5908--5915},
	file = {Sun et al. - 2020 - Revisiting Probability Distribution Assumptions fo.pdf:/Users/jacquelinemaasch/Zotero/storage/XA3IK9LD/Sun et al. - 2020 - Revisiting Probability Distribution Assumptions fo.pdf:application/pdf},
}

@article{mickey_impact_1989,
	title = {{THE} {IMPACT} {OF} {CONFOUNDER} {SELECTION} {CRITERIA} {ON} {EFFECT} {ESTIMATION}},
	volume = {129},
	issn = {1476-6256, 0002-9262},
	url = {https://academic.oup.com/aje/article/58697/THE},
	doi = {10.1093/oxfordjournals.aje.a115101},
	language = {en},
	number = {1},
	urldate = {2023-03-08},
	journal = {American Journal of Epidemiology},
	author = {Mickey, Ruth M. and Greenland, Sander},
	month = jan,
	year = {1989},
	pages = {125--137},
	file = {Mickey and Greenland - 1989 - THE IMPACT OF CONFOUNDER SELECTION CRITERIA ON EFF.pdf:/Users/jacquelinemaasch/Zotero/storage/27SVHP6R/Mickey and Greenland - 1989 - THE IMPACT OF CONFOUNDER SELECTION CRITERIA ON EFF.pdf:application/pdf},
}

@article{lee_is_2014,
	title = {Is a {Cutoff} of 10\% {Appropriate} for the {Change}-in-{Estimate} {Criterion} of {Confounder} {Identification}?},
	volume = {24},
	issn = {0917-5040, 1349-9092},
	url = {https://www.jstage.jst.go.jp/article/jea/24/2/24_JE20130062/_article},
	doi = {10.2188/jea.JE20130062},
	abstract = {Background: When using the change-in-estimate criterion, a cutoff of 10\% is commonly used to identify confounders. However, the appropriateness of this cutoff has never been evaluated. This study investigated cutoffs required under different conditions.
Methods: Four simulations were performed to select cutoffs that achieved a signiﬁcance level of 5\% and a power of 80\%, using linear regression and logistic regression. A total of 10 000 simulations were run to obtain the percentage differences of the 4 ﬁtted regression coefﬁcients (with and without adjustment).
Results: In linear regression, larger effect size, larger sample size, and lower standard deviation of the error term led to a lower cutoff point at a 5\% signiﬁcance level. In contrast, larger effect size and a lower exposure–confounder correlation led to a lower cutoff point at 80\% power. In logistic regression, a lower odds ratio and larger sample size led to a lower cutoff point at a 5\% signiﬁcance level, while a lower odds ratio, larger sample size, and lower exposure–confounder correlation yielded a lower cutoff point at 80\% power.
Conclusions: Cutoff points for the change-in-estimate criterion varied according to the effect size of the exposure–outcome relationship, sample size, standard deviation of the regression error, and exposure–confounder correlation.},
	language = {en},
	number = {2},
	urldate = {2023-03-08},
	journal = {Journal of Epidemiology},
	author = {Lee, Paul H.},
	year = {2014},
	pages = {161--167},
	file = {Lee - 2014 - Is a Cutoff of 10% Appropriate for the Change-in-E.pdf:/Users/jacquelinemaasch/Zotero/storage/6VM9EMC5/Lee - 2014 - Is a Cutoff of 10% Appropriate for the Change-in-E.pdf:application/pdf},
}

@article{weng_methods_2009,
	title = {Methods of {Covariate} {Selection}: {Directed} {Acyclic} {Graphs} and the {Change}-in-{Estimate} {Procedure}},
	volume = {169},
	issn = {0002-9262, 1476-6256},
	shorttitle = {Methods of {Covariate} {Selection}},
	url = {https://academic.oup.com/aje/article-lookup/doi/10.1093/aje/kwp035},
	doi = {10.1093/aje/kwp035},
	language = {en},
	number = {10},
	urldate = {2023-03-08},
	journal = {American Journal of Epidemiology},
	author = {Weng, H.-Y. and Hsueh, Y.-H. and Messam, L. L. McV. and Hertz-Picciotto, I.},
	month = may,
	year = {2009},
	pages = {1182--1190},
	file = {Weng et al. - 2009 - Methods of Covariate Selection Directed Acyclic G.pdf:/Users/jacquelinemaasch/Zotero/storage/AEV9VFXP/Weng et al. - 2009 - Methods of Covariate Selection Directed Acyclic G.pdf:application/pdf},
}

@article{song_understanding_2020-1,
	title = {{UNDERSTANDING} {THE} {LIMITATIONS} {OF} {VARIATIONAL} {MUTUAL} {INFORMATION} {ESTIMATORS}},
	abstract = {Variational approaches based on neural networks are showing promise for estimating mutual information (MI) between high dimensional variables. However, they can be difﬁcult to use in practice due to poorly understood bias/variance tradeoffs. We theoretically show that, under some conditions, estimators such as MINE exhibit variance that could grow exponentially with the true amount of underlying MI. We also empirically demonstrate that existing estimators fail to satisfy basic self-consistency properties of MI, such as data processing and additivity under independence. Based on a uniﬁed perspective of variational approaches, we develop a new estimator that focuses on variance reduction. Empirical results on standard benchmark tasks demonstrate that our proposed estimator exhibits improved biasvariance trade-offs on standard benchmark tasks.},
	language = {en},
	author = {Song, Jiaming and Ermon, Stefano},
	year = {2020},
	file = {Song and Ermon - 2020 - UNDERSTANDING THE LIMITATIONS OF VARIATIONAL MUTUA.pdf:/Users/jacquelinemaasch/Zotero/storage/QVRCSEI6/Song and Ermon - 2020 - UNDERSTANDING THE LIMITATIONS OF VARIATIONAL MUTUA.pdf:application/pdf},
}

@article{poole_variational_nodate,
	title = {On {Variational} {Bounds} of {Mutual} {Information}},
	abstract = {Estimating and optimizing Mutual Information (MI) is core to many problems in machine learning; however, bounding MI in high dimensions is challenging. To establish tractable and scalable objectives, recent work has turned to variational bounds parameterized by neural networks, but the relationships and tradeoffs between these bounds remains unclear. In this work, we unify these recent developments in a single framework. We ﬁnd that the existing variational lower bounds degrade when the MI is large, exhibiting either high bias or high variance. To address this problem, we introduce a continuum of lower bounds that encompasses previous bounds and ﬂexibly trades off bias and variance. On high-dimensional, controlled problems, we empirically characterize the bias and variance of the bounds and their gradients and demonstrate the effectiveness of our new bounds for estimation and representation learning.},
	language = {en},
	author = {Poole, Ben and Ozair, Sherjil},
	file = {Poole and Ozair - On Variational Bounds of Mutual Information.pdf:/Users/jacquelinemaasch/Zotero/storage/6AFSQ2IC/Poole and Ozair - On Variational Bounds of Mutual Information.pdf:application/pdf},
}

@inproceedings{compton_entropic_2020,
	address = {Vancouver, Canada},
	title = {Entropic {Causal} {Inference}: {Identiﬁability} and {Finite} {Sample} {Results}},
	abstract = {Entropic causal inference is a framework for inferring the causal direction between two categorical variables from observational data. The central assumption is that the amount of unobserved randomness in the system is not too large. This unobserved randomness is measured by the entropy of the exogenous variable in the underlying structural causal model, which governs the causal relation between the observed variables. [15] conjectured that the causal direction is identiﬁable when the entropy of the exogenous variable is not too large. In this paper, we prove a variant of their conjecture. Namely, we show that for almost all causal models where the exogenous variable has entropy that does not scale with the number of states of the observed variables, the causal direction is identiﬁable from observational data. We also consider the minimum entropy coupling-based algorithmic approach presented by [15], and for the ﬁrst time demonstrate algorithmic identiﬁability guarantees using a ﬁnite number of samples. We conduct extensive experiments to evaluate the robustness of the method to relaxing some of the assumptions in our theory and demonstrate that both the constant-entropy exogenous variable and the no latent confounder assumptions can be relaxed in practice. We also empirically characterize the number of observational samples needed for causal identiﬁcation. Finally, we apply the algorithm on Tübingen cause-effect pairs dataset.},
	language = {en},
	booktitle = {Proceedings of the 34th {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Compton, Spencer and Kocaoglu, Murat and Greenewald, Kristjan and Katz, Dmitriy},
	year = {2020},
	file = {Compton et al. - Entropic Causal Inference Identiﬁability and Fini.pdf:/Users/jacquelinemaasch/Zotero/storage/UECBZNKH/Compton et al. - Entropic Causal Inference Identiﬁability and Fini.pdf:application/pdf},
}

@article{compton_entropic_nodate,
	title = {Entropic {Causal} {Inference}: {Graph} {Identifiability}},
	abstract = {Entropic causal inference is a recent framework for learning the causal graph between two variables from observational data by finding the information-theoretically simplest structural explanation of the data, i.e., the model with smallest entropy. In our work, we first extend the causal graph identifiability result in the two-variable setting under relaxed assumptions. We then show the first identifiability result using the entropic approach for learning causal graphs with more than two nodes. Our approach utilizes the property that ancestrality between a source node and its descendants can be determined using the bivariate entropic tests. We provide a sound sequential peeling algorithm for general graphs that relies on this property. We also propose a heuristic algorithm for small graphs that shows strong empirical performance. We rigorously evaluate the performance of our algorithms on synthetic data generated from a variety of models, observing improvement over prior work. Finally we test our algorithms on real-world datasets.},
	language = {en},
	author = {Compton, Spencer and Greenewald, Kristjan and Katz, Dmitriy and Kocaoglu, Murat},
	file = {Compton et al. - Entropic Causal Inference Graph Identifiability.pdf:/Users/jacquelinemaasch/Zotero/storage/RWKG8B5H/Compton et al. - Entropic Causal Inference Graph Identifiability.pdf:application/pdf},
}

@misc{ghassami_interaction_2017,
	title = {Interaction {Information} for {Causal} {Inference}: {The} {Case} of {Directed} {Triangle}},
	shorttitle = {Interaction {Information} for {Causal} {Inference}},
	url = {http://arxiv.org/abs/1701.08868},
	abstract = {To be considered for the 2017 IEEE Jack Keil Wolf ISIT Student Paper Award. - Interaction information is one of the multivariate generalizations of mutual information, which expresses the amount information shared among a set of variables, beyond the information, which is shared in any proper subset of those variables. Unlike (conditional) mutual information, which is always non-negative, interaction information can be negative. We utilize this property to ﬁnd the direction of causal inﬂuences among variables in a triangle topology under some mild assumptions.},
	language = {en},
	urldate = {2023-03-10},
	publisher = {arXiv},
	author = {Ghassami, AmirEmad and Kiyavash, Negar},
	month = jan,
	year = {2017},
	note = {arXiv:1701.08868 [cs]},
	keywords = {Computer Science - Artificial Intelligence},
	file = {Ghassami and Kiyavash - 2017 - Interaction Information for Causal Inference The .pdf:/Users/jacquelinemaasch/Zotero/storage/AYV8D2XM/Ghassami and Kiyavash - 2017 - Interaction Information for Causal Inference The .pdf:application/pdf},
}

@article{yeung_new_1991,
	title = {A new outlook on {Shannon}'s information measures},
	volume = {37},
	issn = {0018-9448},
	url = {http://ieeexplore.ieee.org/document/79902/},
	doi = {10.1109/18.79902},
	language = {en},
	number = {3},
	urldate = {2023-03-13},
	journal = {IEEE Transactions on Information Theory},
	author = {Yeung, R.W.},
	month = may,
	year = {1991},
	pages = {466--474},
	file = {Yeung - 1991 - A new outlook on Shannon's information measures.pdf:/Users/jacquelinemaasch/Zotero/storage/JT4CXDZA/Yeung - 1991 - A new outlook on Shannon's information measures.pdf:application/pdf},
}

@article{tsujishita_triple_1995,
	title = {On triple mutual information},
	volume = {16},
	journal = {Advances in Applied Mathematics},
	author = {Tsujishita, Toru},
	year = {1995},
	pages = {269--274},
	file = {Tsujishita - 1995 - On triple mutual information.pdf:/Users/jacquelinemaasch/Zotero/storage/FS478H2G/Tsujishita - 1995 - On triple mutual information.pdf:application/pdf},
}

@article{schamberg_direct_2020,
	title = {Direct and {Indirect} {Effects}—{An} {Information} {Theoretic} {Perspective}},
	volume = {22},
	issn = {1099-4300},
	url = {https://www.mdpi.com/1099-4300/22/8/854},
	doi = {10.3390/e22080854},
	abstract = {Information theoretic (IT) approaches to quantifying causal inﬂuences have experienced some popularity in the literature, in both theoretical and applied (e.g., neuroscience and climate science) domains. While these causal measures are desirable in that they are model agnostic and can capture non-linear interactions, they are fundamentally different from common statistical notions of causal inﬂuence in that they (1) compare distributions over the effect rather than values of the effect and (2) are deﬁned with respect to random variables representing a cause rather than speciﬁc values of a cause. We here present IT measures of direct, indirect, and total causal effects. The proposed measures are unlike existing IT techniques in that they enable measuring causal effects that are deﬁned with respect to speciﬁc values of a cause while still offering the ﬂexibility and general applicability of IT techniques. We provide an identiﬁability result and demonstrate application of the proposed measures in estimating the causal effect of the El Niño–Southern Oscillation on temperature anomalies in the North American Paciﬁc Northwest.},
	language = {en},
	number = {8},
	urldate = {2023-04-01},
	journal = {Entropy},
	author = {Schamberg, Gabriel and Chapman, William and Xie, Shang-Ping and Coleman, Todd},
	month = jul,
	year = {2020},
	pages = {854},
	file = {Schamberg et al. - 2020 - Direct and Indirect Effects—An Information Theoret.pdf:/Users/jacquelinemaasch/Zotero/storage/TJMUPBSB/Schamberg et al. - 2020 - Direct and Indirect Effects—An Information Theoret.pdf:application/pdf},
}

@article{schisterman_overadjustment_2009,
	title = {Overadjustment {Bias} and {Unnecessary} {Adjustment} in {Epidemiologic} {Studies}},
	volume = {20},
	issn = {1044-3983},
	url = {https://journals.lww.com/00001648-200907000-00004},
	doi = {10.1097/EDE.0b013e3181a819a1},
	abstract = {Overadjustment is defined inconsistently. This term is meant to describe control (eg, by regression adjustment, stratification, or restriction) for a variable that either increases net bias or decreases precision without affecting bias. We define overadjustment bias as control for an intermediate variable (or a descending proxy for an intermediate variable) on a causal path from exposure to outcome. We define unnecessary adjustment as control for a variable that does not affect bias of the causal relation between exposure and outcome but may affect its precision. We use causal diagrams and an empirical example (the effect of maternal smoking on neonatal mortality) to illustrate and clarify the definition of overadjustment bias, and to distinguish overadjustment bias from unnecessary adjustment. Using simulations, we quantify the amount of bias associated with overadjustment. Moreover, we show that this bias is based on a different causal structure from confounding or selection biases. Overadjustment bias is not a finite sample bias, while inefficiencies due to control for unnecessary variables are a function of sample size.},
	language = {en},
	number = {4},
	urldate = {2023-04-13},
	journal = {Epidemiology},
	author = {Schisterman, Enrique F. and Cole, Stephen R. and Platt, Robert W.},
	month = jul,
	year = {2009},
	pages = {488--495},
	file = {Schisterman et al. - 2009 - Overadjustment Bias and Unnecessary Adjustment in .pdf:/Users/jacquelinemaasch/Zotero/storage/3XWFN2VY/Schisterman et al. - 2009 - Overadjustment Bias and Unnecessary Adjustment in .pdf:application/pdf},
}

@article{van_zwieten_avoiding_2022,
	title = {Avoiding overadjustment bias in social epidemiology through appropriate covariate selection: a primer},
	volume = {149},
	issn = {08954356},
	shorttitle = {Avoiding overadjustment bias in social epidemiology through appropriate covariate selection},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0895435622001433},
	doi = {10.1016/j.jclinepi.2022.05.021},
	abstract = {Obtaining accurate estimates of the causal effects of socioeconomic position (SEP) on health is important for public health interventions. To do this, researchers must identify and adjust for all potential confounding variables, while avoiding inappropriate adjustment for mediator variables on a causal pathway between the exposure and outcome. Unfortunately, ‘overadjustment bias’ remains a common and under-recognized problem in social epidemiology. This paper offers an introduction on selecting appropriate variables for adjustment when examining effects of SEP on health, with a focus on overadjustment bias. We discuss the challenges of estimating different causal effects including overadjustment bias, provide guidance on overcoming them, and consider speciﬁc issues including the timing of variables across the life-course, mutual adjustment for socioeconomic indicators, and conducting systematic reviews. We recommend three key steps to select the most appropriate variables for adjustment. First, researchers should be clear about their research question and causal effect of interest. Second, using expert knowledge and theory, researchers should draw causal diagrams representing their assumptions about the interrelationships between their variables of interest. Third, based on their causal diagram(s) and causal effect(s) of interest, researchers should select the most appropriate set of variables, which maximizes adjustment for confounding while minimizing adjustment for mediators. Ó 2022 Elsevier Inc. All rights reserved.},
	language = {en},
	urldate = {2023-04-13},
	journal = {Journal of Clinical Epidemiology},
	author = {van Zwieten, Anita and Tennant, Peter W.G. and Kelly-Irving, Michelle and Blyth, Fiona M. and Teixeira-Pinto, Armando and Khalatbari-Soltani, Saman},
	month = sep,
	year = {2022},
	pages = {127--136},
	file = {van Zwieten et al. - 2022 - Avoiding overadjustment bias in social epidemiolog.pdf:/Users/jacquelinemaasch/Zotero/storage/U8VLE92A/van Zwieten et al. - 2022 - Avoiding overadjustment bias in social epidemiolog.pdf:application/pdf},
}

@article{lu_revisiting_2021,
	title = {Revisiting {Overadjustment} {Bias}},
	volume = {32},
	issn = {1044-3983},
	url = {https://journals.lww.com/10.1097/EDE.0000000000001377},
	doi = {10.1097/EDE.0000000000001377},
	language = {en},
	number = {5},
	urldate = {2023-04-13},
	journal = {Epidemiology},
	author = {Lu, Haidong and Cole, Stephen R. and Platt, Robert W. and Schisterman, Enrique F.},
	month = sep,
	year = {2021},
	pages = {e22--e23},
	file = {Lu et al. - 2021 - Revisiting Overadjustment Bias.pdf:/Users/jacquelinemaasch/Zotero/storage/NNIYLDRF/Lu et al. - 2021 - Revisiting Overadjustment Bias.pdf:application/pdf},
}

@article{holmberg_collider_2022,
	title = {Collider bias},
	volume = {327},
	number = {13},
	journal = {JAMA Guide to Statistics and Methods},
	author = {Holmberg, Mathias J. and Andersen, Lars W.},
	year = {2022},
	file = {Holmberg and Andersen - 2022 - Collider bias.pdf:/Users/jacquelinemaasch/Zotero/storage/D3QGPCKV/Holmberg and Andersen - 2022 - Collider bias.pdf:application/pdf},
}

@article{rijnhart_mediation_2021,
	title = {Mediation analysis methods used in observational research: a scoping review and recommendations},
	volume = {21},
	issn = {1471-2288},
	shorttitle = {Mediation analysis methods used in observational research},
	url = {https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-021-01426-3},
	doi = {10.1186/s12874-021-01426-3},
	abstract = {Background:  Mediation analysis methodology underwent many advancements throughout the years, with the most recent and important advancement being the development of causal mediation analysis based on the counterfactual framework. However, a previous review showed that for experimental studies the uptake of causal mediation analysis remains low. The aim of this paper is to review the methodological characteristics of mediation analyses performed in observational epidemiologic studies published between 2015 and 2019 and to provide recommendations for the application of mediation analysis in future studies.
Methods:  We searched the MEDLINE and EMBASE databases for observational epidemiologic studies published between 2015 and 2019 in which mediation analysis was applied as one of the primary analysis methods. Information was extracted on the characteristics of the mediation model and the applied mediation analysis method.
Results:  We included 174 studies, most of which applied traditional mediation analysis methods (n = 123, 70.7\%). Causal mediation analysis was not often used to analyze more complicated mediation models, such as multiple mediator models. Most studies adjusted their analyses for measured confounders, but did not perform sensitivity analyses for unmeasured confounders and did not assess the presence of an exposure-mediator interaction.
Conclusions:  To ensure a causal interpretation of the effect estimates in the mediation model, we recommend that researchers use causal mediation analysis and assess the plausibility of the causal assumptions. The uptake of causal mediation analysis can be enhanced through tutorial papers that demonstrate the application of causal mediation analysis, and through the development of software packages that facilitate the causal mediation analysis of relatively complicated mediation models.},
	language = {en},
	number = {1},
	urldate = {2023-04-18},
	journal = {BMC Medical Research Methodology},
	author = {Rijnhart, Judith J. M. and Lamp, Sophia J. and Valente, Matthew J. and MacKinnon, David P. and Twisk, Jos W. R. and Heymans, Martijn W.},
	month = oct,
	year = {2021},
	pages = {226},
	file = {Rijnhart et al. - 2021 - Mediation analysis methods used in observational r.pdf:/Users/jacquelinemaasch/Zotero/storage/7F8HSM34/Rijnhart et al. - 2021 - Mediation analysis methods used in observational r.pdf:application/pdf},
}

@article{zhao_part_2016,
	title = {Part mutual information for quantifying direct associations in networks},
	volume = {113},
	issn = {0027-8424, 1091-6490},
	url = {https://pnas.org/doi/full/10.1073/pnas.1522586113},
	doi = {10.1073/pnas.1522586113},
	abstract = {Significance
            Measuring direct associations between variables is of great importance in various areas of science, especially in the era of big data. Although mutual information and conditional mutual information are widely used in quantifying both linear and nonlinear associations, they suffer from the serious problems of overestimation and underestimation. To overcome these problems, in contrast to conditional independence, we propose a novel concept of “partial independence” with a new measure, “part mutual information,” based on information theory that can accurately quantify the nonlinearly direct associations between the measured variables.
          , 
            
              Quantitatively identifying direct dependencies between variables is an important task in data analysis, in particular for reconstructing various types of networks and causal relations in science and engineering. One of the most widely used criteria is partial correlation, but it can only measure linearly direct association and miss nonlinear associations. However, based on conditional independence, conditional mutual information (CMI) is able to quantify nonlinearly direct relationships among variables from the observed data, superior to linear measures, but suffers from a serious problem of underestimation, in particular for those variables with tight associations in a network, which severely limits its applications. In this work, we propose a new concept, “partial independence,” with a new measure, “part mutual information” (PMI), which not only can overcome the problem of CMI but also retains the quantification properties of both mutual information (MI) and CMI. Specifically, we first defined PMI to measure nonlinearly direct dependencies between variables and then derived its relations with MI and CMI. Finally, we used a number of simulated data as benchmark examples to numerically demonstrate PMI features and further real gene expression data from
              Escherichia coli
              and yeast to reconstruct gene regulatory networks, which all validated the advantages of PMI for accurately quantifying nonlinearly direct associations in networks.},
	language = {en},
	number = {18},
	urldate = {2023-04-18},
	journal = {Proceedings of the National Academy of Sciences},
	author = {Zhao, Juan and Zhou, Yiwei and Zhang, Xiujun and Chen, Luonan},
	month = may,
	year = {2016},
	pages = {5130--5135},
	file = {Zhao et al. - 2016 - Part mutual information for quantifying direct ass.pdf:/Users/jacquelinemaasch/Zotero/storage/L4MULQPS/Zhao et al. - 2016 - Part mutual information for quantifying direct ass.pdf:application/pdf},
}

@article{pearl_interpretation_2014,
	title = {Interpretation and identification of causal mediation.},
	volume = {19},
	issn = {1939-1463, 1082-989X},
	url = {http://doi.apa.org/getdoi.cfm?doi=10.1037/a0036434},
	doi = {10.1037/a0036434},
	abstract = {This paper reviews the foundations of causal mediation analysis and oﬀers a general and transparent account of the conditions necessary for the identiﬁcation of natural direct and indirect eﬀects, thus facilitating a more informed judgment of the plausibility of these conditions in speciﬁc applications. We show that the conditions usually cited in the literature are overly restrictive, and can be relaxed substantially, without compromising identiﬁcation. In particular, we show that natural eﬀects can be identiﬁed by methods that go beyond standard adjustment for confounders, applicable to observational studies in which treatment assignment remains confounded with the mediator or with the outcome. These identiﬁcation conditions can be validated algorithmically from the diagramatic description of one’s model, and are guaranteed to produce unbiased results whenever the description is correct. The identiﬁcation conditions can be further relaxed in parametric models, possibly including interactions, and permit us to compare the relative importance of several pathways, mediated by interdependent variables.},
	language = {en},
	number = {4},
	urldate = {2023-04-18},
	journal = {Psychological Methods},
	author = {Pearl, Judea},
	year = {2014},
	pages = {459--481},
	file = {Pearl - 2014 - Interpretation and identification of causal mediat.pdf:/Users/jacquelinemaasch/Zotero/storage/6FU7L487/Pearl - 2014 - Interpretation and identification of causal mediat.pdf:application/pdf},
}

@article{carter_information-based_2020,
	title = {An {Information}-{Based} {Approach} for {Mediation} {Analysis} on {High}-{Dimensional} {Metagenomic} {Data}},
	volume = {11},
	issn = {1664-8021},
	url = {https://www.frontiersin.org/article/10.3389/fgene.2020.00148/full},
	doi = {10.3389/fgene.2020.00148},
	language = {en},
	urldate = {2023-04-18},
	journal = {Frontiers in Genetics},
	author = {Carter, Kyle M. and Lu, Meng and Jiang, Hongmei and An, Lingling},
	month = mar,
	year = {2020},
	pages = {148},
	file = {Carter et al. - 2020 - An Information-Based Approach for Mediation Analys.pdf:/Users/jacquelinemaasch/Zotero/storage/TMGV6V4X/Carter et al. - 2020 - An Information-Based Approach for Mediation Analys.pdf:application/pdf},
}

@misc{janzing_justifying_2014,
	title = {Justifying {Information}-{Geometric} {Causal} {Inference}},
	url = {http://arxiv.org/abs/1402.2499},
	abstract = {Information Geometric Causal Inference (IGCI) is a new approach to distinguish between cause and effect for two variables. It is based on an independence assumption between input distribution and causal mechanism that can be phrased in terms of orthogonality in information space. We describe two intuitive reinterpretations of this approach that makes IGCI more accessible to a broader audience. Moreover, we show that the described independence is related to the hypothesis that unsupervised learning and semi-supervised learning only works for predicting the cause from the effect and not vice versa.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Janzing, Dominik and Steudel, Bastian and Shajarisales, Naji and Schölkopf, Bernhard},
	month = feb,
	year = {2014},
	note = {arXiv:1402.2499 [stat]},
	keywords = {Statistics - Machine Learning},
	file = {Janzing et al. - 2014 - Justifying Information-Geometric Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/YK3PBLD8/Janzing et al. - 2014 - Justifying Information-Geometric Causal Inference.pdf:application/pdf},
}

@article{heckerman_learning_nodate,
	title = {Learning {Bayesian} {Networks}: {The} {Combination} of {Knowledge} and {Statistical} {Data}},
	abstract = {We describe a Bayesian approach for learning Bayesian networks from a combination of prior knowledge and statistical data. First and foremost, we develop a methodology for assessing informative priors needed for learning. Our approach is derived from a set of assumptions made previously as well as the assumption of likelihood equivalence, which says that data should not help to discriminate network structures that represent the same assertions of conditional independence. We show that likelihood equivalence when combined with previously made assumptions implies that the user's priors for network parameters can be encoded in a single Bayesian network for the next case to be seen—a prior network—and a single measure of confidence for that network. Second, using these priors, we show how to compute the relative posterior probabilities of network structures given data. Third, we describe search methods for identifying network structures with high posterior probabilities. We describe polynomial algorithms for finding the highest-scoring network structures in the special case where every node has at most k = 1 parent. For the general case (k {\textgreater} 1), which is NP-hard, we review heuristic search algorithms including local search, iterative local search, and simulated annealing. Finally, we describe a methodology for evaluating Bayesian-network learning algorithms, and apply this approach to a comparison of various approaches.},
	language = {en},
	author = {Heckerman, David},
	file = {Heckerman - Learning Bayesian Networks The Combination of Kno.pdf:/Users/jacquelinemaasch/Zotero/storage/NBU5IXZR/Heckerman - Learning Bayesian Networks The Combination of Kno.pdf:application/pdf},
}

@article{cao_posterior_2019,
	title = {Posterior graph selection and estimation consistency for high-dimensional {Bayesian} {DAG} models},
	volume = {47},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-47/issue-1/Posterior-graph-selection-and-estimation-consistency-for-high-dimensional-Bayesian/10.1214/18-AOS1689.full},
	doi = {10.1214/18-AOS1689},
	language = {en},
	number = {1},
	urldate = {2023-04-18},
	journal = {The Annals of Statistics},
	author = {Cao, Xuan and Khare, Kshitij and Ghosh, Malay},
	month = feb,
	year = {2019},
	file = {Cao et al. - 2019 - Posterior graph selection and estimation consisten.pdf:/Users/jacquelinemaasch/Zotero/storage/V9MTWEDF/Cao et al. - 2019 - Posterior graph selection and estimation consisten.pdf:application/pdf},
}
@inproceedings{ghoshal2018learning,
  title={Learning linear structural equation models in polynomial time and sample complexity},
  author={Ghoshal, Asish and Honorio, Jean},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={1466--1475},
  year={2018},
  organization={PMLR}
}
@article{peluso_compatible_2020,
	title = {Compatible priors for model selection of high-dimensional {Gaussian} {DAGs}},
	volume = {14},
	issn = {1935-7524},
	url = {https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-14/issue-2/Compatible-priors-for-model-selection-of-high-dimensional-Gaussian-DAGs/10.1214/20-EJS1768.full},
	doi = {10.1214/20-EJS1768},
	abstract = {Graphical models represent a powerful framework to incorporate conditional independence structure for the statistical analysis of high-dimensional data. In this paper we focus on Directed Acyclic Graphs (DAGs). In the Gaussian setting, a prior recently introduced for the parameters associated to the (modiﬁed) Cholesky decomposition of the precision matrix is the DAG-Wishart. The ﬂexibility introduced through a rich choice of shape hyperparameters coupled with conjugacy are two desirable assets of this prior which are especially welcome for estimation and prediction. In this paper we look at the DAG-Wishart prior from the perspective of model selection, with special reference to its consistency properties in high dimensional settings. We show that Bayes factor consistency only holds when comparing two DAGs which do not belong to the same Markov equivalence class, equivalently they encode distinct conditional independencies; a similar result holds for posterior ratio consistency. We also prove that DAG-Wishart distributions with arbitrarily chosen hyperparameters will lead to incompatible priors for model selection, because they assign different marginal likelihoods to Markov equivalent graphs. To overcome this diﬃculty, we propose a constructive method to specify DAG-Wishart priors whose suitably constrained shape hyperparameters ensure compatibility for DAG model selection.},
	language = {en},
	number = {2},
	urldate = {2023-04-18},
	journal = {Electronic Journal of Statistics},
	author = {Peluso, Stefano and Consonni, Guido},
	month = jan,
	year = {2020},
	file = {Peluso and Consonni - 2020 - Compatible priors for model selection of high-dime.pdf:/Users/jacquelinemaasch/Zotero/storage/I6WHKRLD/Peluso and Consonni - 2020 - Compatible priors for model selection of high-dime.pdf:application/pdf},
}

@article{cooper_bayesian_1992,
	title = {A {Bayesian} method for the induction of probabilistic networks from data},
	volume = {9},
	issn = {0885-6125, 1573-0565},
	url = {http://link.springer.com/10.1007/BF00994110},
	doi = {10.1007/BF00994110},
	abstract = {This paper presents a Bayesianmethod for constructingprobabilistic networksfrom databases. In particular,we focuson constructingBayesianbelief networks.Potentialapplicationsincludecomputer-assistedhypothesis testing,automatedscientificdiscovery,and automatedconstructionof probabifisticexpert systems.We extend the basic method to handle missing data and hidden (latent) variables. We show how to perform probahilistic inference by averagingover the inferences of multiple belief networks. Results are presented of a preliminary evaluationof an algorithmfor constructinga beliefnetworkfrom a databaseof cases. Finally,we relatethe methods in this paper to previous work, and we discuss open problems.},
	language = {en},
	number = {4},
	urldate = {2023-04-18},
	journal = {Machine Learning},
	author = {Cooper, Gregory F. and Herskovits, Edward},
	month = oct,
	year = {1992},
	pages = {309--347},
	file = {Cooper and Herskovits - 1992 - A Bayesian method for the induction of probabilist.pdf:/Users/jacquelinemaasch/Zotero/storage/WRTAZB49/Cooper and Herskovits - 1992 - A Bayesian method for the induction of probabilist.pdf:application/pdf},
}

@article{castelletti_bayesian_2020-1,
	title = {Bayesian {Model} {Selection} of {Gaussian} {Directed} {Acyclic} {Graph} {Structures}},
	volume = {88},
	issn = {0306-7734, 1751-5823},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/insr.12379},
	doi = {10.1111/insr.12379},
	abstract = {During the last years, graphical models have become a popular tool to represent dependencies among variables in many scientiﬁc areas. Typically, the objective is to discover dependence relationships that can be represented through a directed acyclic graph (DAG). The set of all conditional independencies encoded by a DAG determines its Markov property. In general, DAGs encoding the same conditional independencies are not distinguishable from observational data and can be collected into equivalence classes, each one represented by a chain graph called essential graph (EG). However, both the DAG and EG space grow super exponentially in the number of variables, and so, graph structural learning requires the adoption of Markov chain Monte Carlo (MCMC) techniques. In this paper, we review some recent results on Bayesian model selection of Gaussian DAG models under a uniﬁed framework. These results are based on closed-form expressions for the marginal likelihood of a DAG and EG structure, which is obtained from a few suitable assumptions on the prior for model parameters. We then introduce a general MCMC scheme that can be adopted both for model selection of DAGs and EGs together with a couple of applications on real data sets.},
	language = {en},
	number = {3},
	urldate = {2023-04-18},
	journal = {International Statistical Review},
	author = {Castelletti, Federico},
	month = dec,
	year = {2020},
	pages = {752--775},
	file = {Castelletti - 2020 - Bayesian Model Selection of Gaussian Directed Acyc.pdf:/Users/jacquelinemaasch/Zotero/storage/5QEAJA6B/Castelletti - 2020 - Bayesian Model Selection of Gaussian Directed Acyc.pdf:application/pdf},
}

@misc{ben-david_high_2015,
	title = {High dimensional {Bayesian} inference for {Gaussian} directed acyclic graph models},
	url = {http://arxiv.org/abs/1109.4371},
	abstract = {In this paper, we consider Gaussian models Markov with respect to an arbitrary DAG. We ﬁrst construct a family of conjugate priors for the Cholesky parametrization of the covariance matrix of such models. This family has as many shape parameters as the DAG has vertices, and naturally extends the work of Geiger and Heckerman [8]. From these distributions, we derive prior distributions for the covariance and precision parameters of the Gaussian DAG Markov models. Our works thus extends the work of Dawid and Lauritzen [5] and Letac and Massam [16] for Gaussian models Markov with respect to a decomposable graph to arbitrary DAGs. For this reason, we call our distributions DAG-Wishart distributions. An advantage of these distributions is that they possess strong hyper Markov properties and thus allow for explicit estimation of the covariance and precision parameters, regardless of the dimension of the problem. They also allow us to develop methodology for model selection and covariance estimation in the space of DAG-Markov models. We demonstrate via several numerical examples that the proposed method scales well to high-dimensions.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Ben-David, Emanuel and Li, Tianxi and Massam, Helene and Rajaratnam, Bala},
	month = mar,
	year = {2015},
	note = {arXiv:1109.4371 [math, stat]},
	keywords = {Mathematics - Statistics Theory, 62-09, 62E10, 62J05, Statistics - Other Statistics},
	file = {Ben-David et al. - 2015 - High dimensional Bayesian inference for Gaussian d.pdf:/Users/jacquelinemaasch/Zotero/storage/JI62TZ48/Ben-David et al. - 2015 - High dimensional Bayesian inference for Gaussian d.pdf:application/pdf},
}

@article{scheines_bayesian_1999,
	title = {Bayesian estimation and testing of structural equation models},
	volume = {64},
	issn = {0033-3123, 1860-0980},
	url = {http://link.springer.com/10.1007/BF02294318},
	doi = {10.1007/BF02294318},
	abstract = {The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model.},
	language = {en},
	number = {1},
	urldate = {2023-04-18},
	journal = {Psychometrika},
	author = {Scheines, Richard and Hoijtink, Herbert and Boomsma, Anne},
	month = mar,
	year = {1999},
	pages = {37--52},
	file = {Scheines et al. - 1999 - Bayesian estimation and testing of structural equa.pdf:/Users/jacquelinemaasch/Zotero/storage/7B5KVKSU/Scheines et al. - 1999 - Bayesian estimation and testing of structural equa.pdf:application/pdf},
}

@article{kurutmaz_bayesian_nodate,
	title = {{BAYESIAN} {MODEL} {SELECTION} {FOR} {LATENT} {VARIABLE} {CAUSAL} {NETWORKS} {BY} {SEQUENTIAL} {MONTE} {CARLO}},
	language = {en},
	author = {Kurutmaz, Mehmet Burak},
	file = {Kurutmaz - BAYESIAN MODEL SELECTION FOR LATENT VARIABLE CAUSA.pdf:/Users/jacquelinemaasch/Zotero/storage/FGS8I9BP/Kurutmaz - BAYESIAN MODEL SELECTION FOR LATENT VARIABLE CAUSA.pdf:application/pdf},
}

@misc{hasan_survey_2023,
	title = {A {Survey} on {Causal} {Discovery} {Methods} for {Temporal} and {Non}-{Temporal} {Data}},
	url = {http://arxiv.org/abs/2303.15027},
	abstract = {Causal Discovery (CD) is the process of identifying the cause-eﬀect relationships among the variables of a system from data. Over the years, several methods have been developed primarily based on the statistical properties of data to uncover the underlying causal mechanism. In this study, we present an extensive discussion on the methods designed to perform causal discovery from both independent and identically distributed (i.i.d.) data and time series data. For this purpose, we ﬁrst introduce the common terminologies in causal discovery, and then provide a comprehensive discussion of the algorithms designed to identify the causal edges in diﬀerent settings. We further discuss some of the benchmark datasets available for evaluating the performance of the causal discovery methods, available tools or software packages to perform causal discovery readily, and the common metrics used to evaluate these methods. We also test some common causal discovery algorithms on diﬀerent benchmark datasets, and compare their performances. Finally, we conclude by presenting the common challenges involved in causal discovery, and also, discuss the applications of causal discovery in multiple areas of interest.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Hasan, Uzma and Hossain, Emam and Gani, Md Osman},
	month = apr,
	year = {2023},
	note = {arXiv:2303.15027 [cs]},
	keywords = {Computer Science - Artificial Intelligence},
	file = {Hasan et al. - 2023 - A Survey on Causal Discovery Methods for Temporal .pdf:/Users/jacquelinemaasch/Zotero/storage/G9NXNXJG/Hasan et al. - 2023 - A Survey on Causal Discovery Methods for Temporal .pdf:application/pdf},
}

@misc{chicharro_causal_2020,
	title = {Causal learning with sufficient statistics: an information bottleneck approach},
	shorttitle = {Causal learning with sufficient statistics},
	url = {http://arxiv.org/abs/2010.05375},
	abstract = {The inference of causal relationships using observational data from partially observed multivariate systems with hidden variables is a fundamental question in many scientiﬁc domains. Methods extracting causal information from conditional independencies between variables of a system are common tools for this purpose, but are limited in the lack of independencies. To surmount this limitation, we capitalize on the fact that the laws governing the generative mechanisms of a system often result in substructures embodied in the generative functional equation of a variable, which act as suﬃcient statistics for the inﬂuence that other variables have on it. These functional suﬃcient statistics constitute intermediate hidden variables providing new conditional independencies to be tested. We propose to use the Information Bottleneck method, a technique commonly applied for dimensionality reduction, to ﬁnd underlying suﬃcient sets of statistics. Using these statistics we formulate new additional rules of causal orientation that provide causal information not obtainable from standard structure learning algorithms, which exploit only conditional independencies between observable variables. We validate the use of suﬃcient statistics for structure learning both with simulated systems built to contain speciﬁc suﬃcient statistics and with benchmark data from regulatory rules previously and independently proposed to model biological signal transduction networks.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Chicharro, Daniel and Besserve, Michel and Panzeri, Stefano},
	month = oct,
	year = {2020},
	note = {arXiv:2010.05375 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, 94A16, 62D20, 62H22, 62Bxx},
	file = {Chicharro et al. - 2020 - Causal learning with sufficient statistics an inf.pdf:/Users/jacquelinemaasch/Zotero/storage/ASAI8S8Y/Chicharro et al. - 2020 - Causal learning with sufficient statistics an inf.pdf:application/pdf},
}

@misc{hayete_identification_2021,
	title = {Identification of {Latent} {Variables} {From} {Graphical} {Model} {Residuals}},
	url = {http://arxiv.org/abs/2101.02332},
	abstract = {Graph-based causal discovery methods aim to capture conditional independencies consistent with the observed data and differentiate causal relationships from indirect or induced ones. Successful construction of graphical models of data depends on the assumption of causal sufﬁciency: that is, that all confounding variables are measured. When this assumption is not met, learned graphical structures may become arbitrarily incorrect and effects implied by such models may be wrongly attributed, carry the wrong magnitude, or mis-represent direction of correlation. Wide application of graphical models to increasingly less curated "big data" draws renewed attention to the unobserved confounder problem.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Hayete, Boris and Gruber, Fred and Decker, Anna and Yan, Raymond},
	month = jan,
	year = {2021},
	note = {arXiv:2101.02332 [cs, q-bio, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Quantitative Biology - Quantitative Methods},
	file = {Hayete et al. - 2021 - Identification of Latent Variables From Graphical .pdf:/Users/jacquelinemaasch/Zotero/storage/CEEV2H8W/Hayete et al. - 2021 - Identification of Latent Variables From Graphical .pdf:application/pdf},
}

@article{chobtham_hybrid_2023,
	title = {Hybrid {Bayesian} network discovery with latent variables by scoring multiple interventions},
	volume = {37},
	issn = {1384-5810, 1573-756X},
	url = {https://link.springer.com/10.1007/s10618-022-00882-9},
	doi = {10.1007/s10618-022-00882-9},
	abstract = {In Bayesian Networks (BNs), the direction of edges is crucial for causal reasoning and inference. However, Markov equivalence class considerations mean it is not always possible to establish edge orientations, which is why many BN structure learning algorithms cannot orientate all edges from purely observational data. Moreover, latent confounders can lead to false positive edges. Relatively few methods have been proposed to address these issues. In this work, we present the hybrid mFGS-BS (majority rule and Fast Greedy equivalence Search with Bayesian Scoring) algorithm for structure learning from discrete data that involves an observational data set and one or more interventional data sets. The algorithm assumes causal insufﬁciency in the presence of latent variables and produces a Partial Ancestral Graph (PAG). Structure learning relies on a hybrid approach and a novel Bayesian scoring paradigm that calculates the posterior probability of each directed edge being added to the learnt graph. Experimental results based on well-known networks of up to 109 variables and 10 k sample size show that mFGS-BS improves structure learning accuracy relative to the stateof-the-art and it is computationally efﬁcient.},
	language = {en},
	number = {1},
	urldate = {2023-04-18},
	journal = {Data Mining and Knowledge Discovery},
	author = {Chobtham, Kiattikun and Constantinou, Anthony C. and Kitson, Neville K.},
	month = jan,
	year = {2023},
	pages = {476--520},
	file = {Chobtham et al. - 2023 - Hybrid Bayesian network discovery with latent vari.pdf:/Users/jacquelinemaasch/Zotero/storage/4KPGW7HK/Chobtham et al. - 2023 - Hybrid Bayesian network discovery with latent vari.pdf:application/pdf},
}

@article{constantinou_large-scale_2021,
	title = {Large-scale empirical validation of {Bayesian} {Network} structure learning algorithms with noisy data},
	volume = {131},
	issn = {0888613X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X21000025},
	doi = {10.1016/j.ijar.2021.01.001},
	language = {en},
	urldate = {2023-04-18},
	journal = {International Journal of Approximate Reasoning},
	author = {Constantinou, Anthony C. and Liu, Yang and Chobtham, Kiattikun and Guo, Zhigao and Kitson, Neville K.},
	month = apr,
	year = {2021},
	pages = {151--188},
	file = {Constantinou et al. - 2021 - Large-scale empirical validation of Bayesian Netwo.pdf:/Users/jacquelinemaasch/Zotero/storage/2QBLK9DN/Constantinou et al. - 2021 - Large-scale empirical validation of Bayesian Netwo.pdf:application/pdf},
}

@article{hoeting_bayesian_nodate,
	title = {Bayesian {Model} {Averaging}: {A} {Tutorial}},
	abstract = {Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This approach ignores the uncertainty in model selection, leading to over-conﬁdent inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA) provides a coherent mechanism for accounting for this model uncertainty. Several methods for implementing BMA have recently emerged. We discuss these methods and present a number of examples. In these examples, BMA provides improved out-ofsample predictive performance. We also provide a catalogue of currently available BMA software.},
	language = {en},
	author = {Hoeting, Jennifer A and Madigan, David and Raftery, Adrian E and Volinsky, Chris T},
	file = {Hoeting et al. - Bayesian Model Averaging A Tutorial.pdf:/Users/jacquelinemaasch/Zotero/storage/QRFATWZA/Hoeting et al. - Bayesian Model Averaging A Tutorial.pdf:application/pdf},
}

@incollection{ceci_discovery_2017,
	address = {Cham},
	title = {Discovery of {Causal} {Models} that {Contain} {Latent} {Variables} {Through} {Bayesian} {Scoring} of {Independence} {Constraints}},
	volume = {10535},
	isbn = {978-3-319-71245-1 978-3-319-71246-8},
	url = {https://link.springer.com/10.1007/978-3-319-71246-8_9},
	abstract = {Discovering causal structure from observational data in the presence of latent variables remains an active research area. Constraint-based causal discovery algorithms are relatively efficient at discovering such causal models from data using independence tests. Typically, however, they derive and output only one such model. In contrast, Bayesian methods can generate and probabilistically score multiple models, outputting the most probable one; however, they are often computationally infeasible to apply when modeling latent variables. We introduce a hybrid method that derives a Bayesian probability that the set of independence tests associated with a given causal model are jointly correct. Using this constraint-based scoring method, we are able to score multiple causal models, which possibly contain latent variables, and output the most probable one. The structure-discovery performance of the proposed method is compared to an existing constraint-based method (RFCI) using data generated from several previously published Bayesian networks. The structural Hamming distances of the output models improved when using the proposed method compared to RFCI, especially for small sample sizes.},
	language = {en},
	urldate = {2023-04-18},
	booktitle = {Machine {Learning} and {Knowledge} {Discovery} in {Databases}},
	publisher = {Springer International Publishing},
	author = {Jabbari, Fattaneh and Ramsey, Joseph and Spirtes, Peter and Cooper, Gregory},
	editor = {Ceci, Michelangelo and Hollmén, Jaakko and Todorovski, Ljupčo and Vens, Celine and Džeroski, Sašo},
	year = {2017},
	doi = {10.1007/978-3-319-71246-8_9},
	note = {Series Title: Lecture Notes in Computer Science},
	pages = {142--157},
	file = {Jabbari et al. - 2017 - Discovery of Causal Models that Contain Latent Var.pdf:/Users/jacquelinemaasch/Zotero/storage/H77M9U6Y/Jabbari et al. - 2017 - Discovery of Causal Models that Contain Latent Var.pdf:application/pdf},
}

@book{koller_probabilistic_2009,
	address = {Cambridge, MA},
	series = {Adaptive computation and machine learning},
	title = {Probabilistic graphical models: principles and techniques},
	isbn = {978-0-262-01319-2},
	shorttitle = {Probabilistic graphical models},
	language = {en},
	publisher = {MIT Press},
	author = {Koller, Daphne and Friedman, Nir},
	year = {2009},
	keywords = {Bayesian statistical decision theory, Graphic methods, Graphical modeling (Statistics)},
	file = {Koller and Friedman - 2009 - Probabilistic graphical models principles and tec.pdf:/Users/jacquelinemaasch/Zotero/storage/CAPD4AJV/Koller and Friedman - 2009 - Probabilistic graphical models principles and tec.pdf:application/pdf},
}

@article{nguyen_magnitude_2019,
	title = {The magnitude and direction of collider bias for binary variables},
	volume = {8},
	issn = {2161-962X, 2194-9263},
	url = {http://arxiv.org/abs/1609.00606},
	doi = {10.1515/em-2017-0013},
	abstract = {Suppose we are interested in the effect of variable X on variable Y . If X and Y both inﬂuence, or are associated with variables that inﬂuence, a common outcome, called a collider, then conditioning on the collider (or on a variable inﬂuenced by the collider – its “child”) induces a spurious association between X and Y , which is known as collider bias. Characterizing the magnitude and direction of collider bias is crucial for understanding the implications of selection bias and for adjudicating decisions about whether to control for variables that are known to be associated with both exposure and outcome but could be either confounders or colliders. Considering a class of situations where all variables are binary, and where X and Y either are, or are respectively inﬂuenced by, two marginally independent causes of a collider, we derive collider bias that results from (i) conditioning on speciﬁc levels of the collider or its child (on the covariance, risk difference, and in two cases odds ratio, scales), or (ii) linear regression adjustment for, the collider or its child. We also derive simple conditions that determine the sign of such bias.},
	language = {en},
	number = {1},
	urldate = {2023-04-18},
	journal = {Epidemiologic Methods},
	author = {Nguyen, Trang Quynh and Dafoe, Allan and Ogburn, Elizabeth L.},
	month = dec,
	year = {2019},
	note = {arXiv:1609.00606 [math, stat]},
	keywords = {Statistics - Methodology, Mathematics - Statistics Theory},
	pages = {20170013},
	file = {Nguyen et al. - 2019 - The magnitude and direction of collider bias for b.pdf:/Users/jacquelinemaasch/Zotero/storage/JMMK72FA/Nguyen et al. - 2019 - The magnitude and direction of collider bias for b.pdf:application/pdf},
}

@misc{peters_causal_2014,
	title = {Causal {Discovery} with {Continuous} {Additive} {Noise} {Models}},
	abstract = {We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identiﬁable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for ﬁnitely many samples, RESIT (Regression with Subsequent Independence Test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Peters, Jonas and Mooij, Joris and Janzing, Dominik and Schölkopf, Bernhard},
	month = apr,
	year = {2014},
	note = {arXiv:1309.6779 [stat]},
	keywords = {Statistics - Machine Learning},
	file = {Peters et al. - 2014 - Causal Discovery with Continuous Additive Noise Mo.pdf:/Users/jacquelinemaasch/Zotero/storage/ZCFPRJGK/Peters et al. - 2014 - Causal Discovery with Continuous Additive Noise Mo.pdf:application/pdf},
}

@misc{janzing_justifying_2009,
	title = {Justifying additive-noise-model based causal discovery via algorithmic information theory},
	url = {http://arxiv.org/abs/0910.1691},
	abstract = {A recent method for causal discovery is in many cases able to infer whether X causes Y or Y causes X for just two observed variables X and Y . It is based on the observation that there exist (non-Gaussian) joint distributions P (X, Y ) for which Y may be written as a function of X up to an additive noise term that is independent of X and no such model exists from Y to X. Whenever this is the case, one prefers the causal model X → Y .},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Janzing, Dominik and Steudel, Bastian},
	month = oct,
	year = {2009},
	note = {arXiv:0910.1691 [cs, math]},
	keywords = {Computer Science - Information Theory},
	file = {Janzing and Steudel - 2009 - Justifying additive-noise-model based causal disco.pdf:/Users/jacquelinemaasch/Zotero/storage/9PQHESQE/Janzing and Steudel - 2009 - Justifying additive-noise-model based causal disco.pdf:application/pdf},
}

@misc{janzing_quantifying_2021,
	title = {Quantifying intrinsic causal contributions via structure preserving interventions},
	url = {http://arxiv.org/abs/2007.00714},
	abstract = {We propose a new notion of causal contribution which describes the ’intrinsic’ part of the contribution of a node on a target node in a DAG. We show that in some scenarios the existing causal quantiﬁcation methods failed to capture this notion exactly. By recursively writing each node as a function of the upstream noise terms, we separate the intrinsic information added by each node from the one obtained from its ancestors. To interpret the intrinsic information as a causal contribution, we consider ’structure-preserving interventions’ that randomize each node in a way that mimics the usual dependence on the parents and do not perturb the observed joint distribution. To get a measure that is invariant across arbitrary orderings of nodes we propose Shapley based symmetrization. We describe our contribution analysis for variance and entropy, but contributions for other target metrics can be deﬁned analogously.},
	language = {en},
	urldate = {2023-04-18},
	publisher = {arXiv},
	author = {Janzing, Dominik and Blöbaum, Patrick and Minorics, Lenon and Faller, Philipp and Mastakouri, Atalanti},
	month = nov,
	year = {2021},
	note = {arXiv:2007.00714 [cs, math, stat]},
	keywords = {Statistics - Machine Learning, Computer Science - Artificial Intelligence, Computer Science - Information Theory},
	file = {Janzing et al. - 2021 - Quantifying intrinsic causal contributions via str.pdf:/Users/jacquelinemaasch/Zotero/storage/DP26EN6M/Janzing et al. - 2021 - Quantifying intrinsic causal contributions via str.pdf:application/pdf},
}

@article{claassen_bayesian_nodate,
	title = {Bayesian {Probabilities} for {Constraint}-based {Causal} {Discovery}},
	abstract = {We target the problem of accuracy and robustness in causal inference from ﬁnite data sets. Our aim is to combine the inherent robustness of the Bayesian approach with the theoretical strength and clarity of constraint-based methods. We use a Bayesian score to obtain probability estimates on the input statements used in a constraint-based procedure. These are subsequently processed in decreasing order of reliability, letting more reliable decisions take precedence in case of conﬂicts, until a single output model is obtained. Tests show that a basic implementation of the resulting Bayesian Constraint-based Causal Discovery (BCCD) algorithm already outperforms established procedures such as FCI and Conservative PC. It indicates which causal decisions in the output have high reliability and which do not. The approach is easily adapted to other application areas such as complex independence tests.},
	language = {en},
	author = {Claassen, Tom and Heskes, Tom},
	file = {Claassen and Heskes - Bayesian Probabilities for Constraint-based Causal.pdf:/Users/jacquelinemaasch/Zotero/storage/HQYBB6U9/Claassen and Heskes - Bayesian Probabilities for Constraint-based Causal.pdf:application/pdf},
}

@article{claassen_bayesian_nodate-1,
	title = {A {Bayesian} {Approach} to {Constraint} {Based} {Causal} {Inference}},
	abstract = {We target the problem of accuracy and robustness in causal inference from ﬁnite data sets. Some state-of-the-art algorithms produce clear output complete with solid theoretical guarantees but are susceptible to propagating erroneous decisions, while others are very adept at handling and representing uncertainty, but need to rely on undesirable assumptions. Our aim is to combine the inherent robustness of the Bayesian approach with the theoretical strength and clarity of constraint-based methods. We use a Bayesian score to obtain probability estimates on the input statements used in a constraint-based procedure. These are subsequently processed in decreasing order of reliability, letting more reliable decisions take precedence in case of conﬂicts, until a single output model is obtained. Tests show that a basic implementation of the resulting Bayesian Constraint-based Causal Discovery (BCCD) algorithm already outperforms established procedures such as FCI and Conservative PC. It can also indicate which causal decisions in the output have high reliability and which do not.},
	language = {en},
	author = {Claassen, Tom and Heskes, Tom},
	file = {Claassen and Heskes - A Bayesian Approach to Constraint Based Causal Inf.pdf:/Users/jacquelinemaasch/Zotero/storage/D7EBBIE3/Claassen and Heskes - A Bayesian Approach to Constraint Based Causal Inf.pdf:application/pdf},
}

@article{austin_introduction_2011,
	title = {An {Introduction} to {Propensity} {Score} {Methods} for {Reducing} the {Effects} of {Confounding} in {Observational} {Studies}},
	volume = {46},
	issn = {0027-3171, 1532-7906},
	url = {http://www.tandfonline.com/doi/abs/10.1080/00273171.2011.568786},
	doi = {10.1080/00273171.2011.568786},
	language = {en},
	number = {3},
	urldate = {2023-04-20},
	journal = {Multivariate Behavioral Research},
	author = {Austin, Peter C.},
	month = may,
	year = {2011},
	pages = {399--424},
	file = {Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:/Users/jacquelinemaasch/Zotero/storage/6JZFVZIK/Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:application/pdf;Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:/Users/jacquelinemaasch/Zotero/storage/NJSJXQC2/Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:application/pdf;Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:/Users/jacquelinemaasch/Zotero/storage/BGZV8DLM/Austin - 2011 - An Introduction to Propensity Score Methods for Re.pdf:application/pdf},
}

@article{cole_fallibility_2002,
	title = {Fallibility in estimating direct effects},
	volume = {31},
	issn = {1464-3685, 0300-5771},
	url = {https://academic.oup.com/ije/article-lookup/doi/10.1093/ije/31.1.163},
	doi = {10.1093/ije/31.1.163},
	language = {en},
	number = {1},
	urldate = {2023-04-24},
	journal = {International Journal of Epidemiology},
	author = {Cole, Stephen R and Hernán, Miguel A},
	month = feb,
	year = {2002},
	pages = {163--165},
	file = {Cole and Hernán - 2002 - Fallibility in estimating direct effects.pdf:/Users/jacquelinemaasch/Zotero/storage/FJ2GEZUE/Cole and Hernán - 2002 - Fallibility in estimating direct effects.pdf:application/pdf},
}

@article{ertefaie_variable_2018,
	title = {Variable {Selection} in {Causal} {Inference} using a {Simultaneous} {Penalization} {Method}},
	volume = {6},
	issn = {2193-3685, 2193-3677},
	url = {https://www.degruyter.com/document/doi/10.1515/jci-2017-0010/html},
	doi = {10.1515/jci-2017-0010},
	abstract = {In the causal adjustment setting, variable selection techniques based only on the outcome or only on the treatment allocation model can result in the omission of confounders and hence may lead to bias, or the inclusion of spurious variables and hence cause variance inflation, in estimation of the treatment effect. We propose a variable selection method using a penalized objective function that is based on both the outcome and treatment assignment models. The proposed method facilitates confounder selection in highdimensional settings. We show that under some mild conditions our method attains the oracle property. The selected variables are used to form a doubly robust regression estimator of the treatment effect. Using the proposed method we analyze a set of data on economic growth and study the effect of life expectancy as a measure of population health on the average growth rate of gross domestic product per capita.},
	language = {en},
	number = {1},
	urldate = {2023-04-24},
	journal = {Journal of Causal Inference},
	author = {Ertefaie, Ashkan and Asgharian, Masoud and Stephens, David A.},
	month = mar,
	year = {2018},
	pages = {20170010},
	file = {Ertefaie et al. - 2018 - Variable Selection in Causal Inference using a Sim.pdf:/Users/jacquelinemaasch/Zotero/storage/7XQ9QIRW/Ertefaie et al. - 2018 - Variable Selection in Causal Inference using a Sim.pdf:application/pdf},
}

@article{koch_variable_2020,
	title = {Variable selection and estimation in causal inference using {Bayesian} spike and slab priors},
	volume = {29},
	issn = {0962-2802, 1477-0334},
	url = {http://journals.sagepub.com/doi/10.1177/0962280219898497},
	doi = {10.1177/0962280219898497},
	abstract = {Unbiased estimation of causal effects with observational data requires adjustment for confounding variables that are related to both the outcome and treatment assignment. Standard variable selection techniques aim to maximize predictive ability of the outcome model, but they ignore covariate associations with treatment and may not adjust for important confounders weakly associated to outcome. We propose a novel method for estimating causal effects that simultaneously considers models for both outcome and treatment, which we call the Bilevel Spike and Slab Causal Estimator (BSSCE). By using a Bayesian formulation, BSSCE estimates the posterior distribution of all model parameters and provides straightforward and reliable inference. Spike and slab priors are used on each covariate coefficient which aim to minimize the mean squared error (MSE) of the treatment effect estimator. Theoretical properties of the treatment effect estimator are derived justifying the prior used in BSSCE. Simulations show BSSCE can substantially reduce MSE over numerous methods, and performs especially well with large numbers of covariates, including situations where the number of covariates is greater than the sample size. We illustrate BSSCE by estimating the causal effect of vasoactive therapy versus fluid resuscitation on hypotensive episode length for patients in the MIMIC-III critical care database.},
	language = {en},
	number = {9},
	urldate = {2023-04-24},
	journal = {Statistical Methods in Medical Research},
	author = {Koch, Brandon and Vock, David M and Wolfson, Julian and Vock, Laura Boehm},
	month = sep,
	year = {2020},
	pages = {2445--2469},
	file = {Koch et al. - 2020 - Variable selection and estimation in causal infere.pdf:/Users/jacquelinemaasch/Zotero/storage/YZHRLUMC/Koch et al. - 2020 - Variable selection and estimation in causal infere.pdf:application/pdf},
}

@article{shortreed_outcomeadaptive_2017,
	title = {Outcome‐adaptive lasso: {Variable} selection for causal inference},
	volume = {73},
	issn = {0006-341X, 1541-0420},
	shorttitle = {Outcome‐adaptive lasso},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12679},
	doi = {10.1111/biom.12679},
	abstract = {Methodological advancements, including propensity score methods, have resulted in improved unbiased estimation of treatment eﬀects from observational data. Traditionally, a “throw in the kitchen sink” approach has been used to select covariates for inclusion into the propensity score, but recent work shows including unnecessary covariates can impact both the bias and statistical eﬃciency of propensity score estimators. In particular, the inclusion of covariates that impact exposure but not the outcome, can inﬂate standard errors without improving bias, while the inclusion of covariates associated with the outcome but unrelated to exposure can improve precision. We propose the outcome-adaptive lasso for selecting appropriate covariates for inclusion in propensity score models to account for confounding bias and maintaining statistical eﬃciency. This proposed approach can perform variable selection in the presence of a large number of spurious covariates, that is, covariates unrelated to outcome or exposure. We present theoretical and simulation results indicating that the outcome-adaptive lasso selects the propensity score model that includes all true confounders and predictors of outcome, while excluding other covariates. We illustrate covariate selection using the outcome-adaptive lasso, including comparison to alternative approaches, using simulated data and in a survey of patients using opioid therapy to manage chronic pain.},
	language = {en},
	number = {4},
	urldate = {2023-04-24},
	journal = {Biometrics},
	author = {Shortreed, Susan M. and Ertefaie, Ashkan},
	month = dec,
	year = {2017},
	pages = {1111--1122},
	file = {Shortreed and Ertefaie - 2017 - Outcome‐adaptive lasso Variable selection for cau.pdf:/Users/jacquelinemaasch/Zotero/storage/LSP7UUA5/Shortreed and Ertefaie - 2017 - Outcome‐adaptive lasso Variable selection for cau.pdf:application/pdf},
}

@article{de_luna_covariate_2011,
	title = {Covariate selection for the nonparametric estimation of an average treatment effect},
	volume = {98},
	issn = {0006-3444, 1464-3510},
	url = {https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/asr041},
	doi = {10.1093/biomet/asr041},
	abstract = {Observational studies in which the effect of a nonrandomized treatment on an outcome of interest is estimated are common in domains such as labour economics and epidemiology. Such studies often rely on an assumption of unconfounded treatment when controlling for a given set of observed pre-treatment covariates. The choice of covariates to control in order to guarantee unconfoundedness should primarily be based on subject matter theories, although the latter typically give only partial guidance. It is tempting to include many covariates in the controlling set to try to make the assumption of an unconfounded treatment realistic. Including unnecessary covariates is suboptimal when the effect of a binary treatment is estimated nonparametrically. For instance, when using a n1/2-consistent estimator, a loss of efficiency may result from using covariates that are irrelevant for the unconfoundedness assumption. Moreover, bias may dominate the variance when many covariates are used. Embracing the Neyman–Rubin model typically used in conjunction with nonparametric estimators of treatment effects, we characterize subsets from the original reservoir of covariates that are minimal in the sense that the treatment ceases to be unconfounded given any proper subset of these minimal sets. These subsets of covariates are shown to be identified under mild assumptions. These results lead us to propose data-driven algorithms for the selection of minimal sets of covariates.},
	language = {en},
	number = {4},
	urldate = {2023-04-24},
	journal = {Biometrika},
	author = {De Luna, X. and Waernbaum, I. and Richardson, T. S.},
	month = dec,
	year = {2011},
	pages = {861--875},
	file = {De Luna et al. - 2011 - Covariate selection for the nonparametric estimati.pdf:/Users/jacquelinemaasch/Zotero/storage/LQZESZYS/De Luna et al. - 2011 - Covariate selection for the nonparametric estimati.pdf:application/pdf},
}

@article{persson_data-driven_2017,
	title = {Data-driven algorithms for dimension reduction in causal inference},
	volume = {105},
	issn = {01679473},
	doi = {10.1016/j.csda.2016.08.012},
	abstract = {In observational studies, the causal effect of a treatment may be confounded with variables that are related to both the treatment and the outcome of interest. In order to identify a causal effect, such studies often rely on the unconfoundedness assumption, i.e., that all confounding variables are observed. The choice of covariates to control for, which is primarily based on subject matter knowledge, may result in a large covariate vector in the attempt to ensure that unconfoundedness holds. However, including redundant covariates can affect bias and efficiency of nonparametric causal effect estimators, e.g., due to the curse of dimensionality. Data-driven algorithms for the selection of sufficient covariate subsets are investigated. Under the assumption of unconfoundedness the algorithms search for minimal subsets of the covariate vector. Based, e.g., on the framework of sufficient dimension reduction or kernel smoothing, the algorithms perform a backward elimination procedure assessing the significance of each covariate. Their performance is evaluated in simulations and an application using data from the Swedish Childhood Diabetes Register is also presented.},
	language = {en},
	urldate = {2023-04-24},
	journal = {Computational Statistics \& Data Analysis},
	author = {Persson, Emma and Häggström, Jenny and Waernbaum, Ingeborg and De Luna, Xavier},
	month = jan,
	year = {2017},
	pages = {280--292},
	file = {Persson et al. - 2017 - Data-driven algorithms for dimension reduction in .pdf:/Users/jacquelinemaasch/Zotero/storage/N5M9FGWP/Persson et al. - 2017 - Data-driven algorithms for dimension reduction in .pdf:application/pdf},
}

@article{belloni_inference_2014,
	title = {Inference on {Treatment} {Effects} after {Selection} among {High}-{Dimensional} {Controls}},
	volume = {81},
	issn = {0034-6527, 1467-937X},
	url = {https://academic.oup.com/restud/article-lookup/doi/10.1093/restud/rdt044},
	doi = {10.1093/restud/rdt044},
	abstract = {We propose robust methods for inference about the effect of a treatment variable on a scalar outcome in the presence of very many regressors in a model with possibly non-Gaussian and heteroscedastic disturbances. We allow for the number of regressors to be larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by including a relatively small number of variables whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of regressors. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the “post-double-selection” method. The main attractive feature of our method is that it allows for imperfect selection of the controls and provides conﬁdence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, ﬁxed number of controls. Thus, our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We also present a generalization of our method to a fully heterogeneous model with a binary treatment variable. We illustrate the use of the developed methods with numerical simulations and an application that considers the effect of abortion on crime rates.},
	language = {en},
	number = {2},
	urldate = {2023-04-24},
	journal = {The Review of Economic Studies},
	author = {Belloni, A. and Chernozhukov, V. and Hansen, C.},
	month = apr,
	year = {2014},
	pages = {608--650},
	file = {Belloni et al. - 2014 - Inference on Treatment Effects after Selection amo.pdf:/Users/jacquelinemaasch/Zotero/storage/JCPQ9W55/Belloni et al. - 2014 - Inference on Treatment Effects after Selection amo.pdf:application/pdf},
}

@article{witte_covariate_2019,
	title = {Covariate selection strategies for causal inference: {Classification} and comparison},
	volume = {61},
	issn = {0323-3847, 1521-4036},
	shorttitle = {Covariate selection strategies for causal inference},
	url = {https://onlinelibrary.wiley.com/doi/10.1002/bimj.201700294},
	doi = {10.1002/bimj.201700294},
	abstract = {When causal eﬀects are to be estimated from observational data, we have to adjust for confounding. A central aim of covariate selection for causal inference is therefore to determine a set that is suﬃcient for confounding adjustment, but other aims such as eﬃciency or robustness can be important as well. In this paper, we review six general approaches to covariate selection that diﬀer in the targeted type of adjustment set. We discuss and illustrate their advantages and disadvantages using causal diagrams. Moreover, the approaches and diﬀerent ways of implementing them are compared empirically in an extensive simulation study. We conclude that there are considerable diﬀerences between the approaches but none of them is uniformly best, with performance depending on the chosen adjustment method as well as the true confounding structure. Any prior structural knowledge on the causal relations is helpful to choose the most appropriate method.},
	language = {en},
	number = {5},
	urldate = {2023-04-24},
	journal = {Biometrical Journal},
	author = {Witte, Janine and Didelez, Vanessa},
	month = sep,
	year = {2019},
	pages = {1270--1289},
	file = {Witte and Didelez - 2019 - Covariate selection strategies for causal inferenc.pdf:/Users/jacquelinemaasch/Zotero/storage/7E2KD6F7/Witte and Didelez - 2019 - Covariate selection strategies for causal inferenc.pdf:application/pdf},
}

@article{maathuis_generalized_2015,
	title = {A generalized back-door criterion},
	volume = {43},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-43/issue-3/A-generalized-back-door-criterion/10.1214/14-AOS1295.full},
	doi = {10.1214/14-AOS1295},
	language = {en},
	number = {3},
	urldate = {2023-04-24},
	journal = {The Annals of Statistics},
	author = {Maathuis, Marloes H. and Colombo, Diego},
	month = jun,
	year = {2015},
	file = {Maathuis and Colombo - 2015 - A generalized back-door criterion.pdf:/Users/jacquelinemaasch/Zotero/storage/3WW5CNCA/Maathuis and Colombo - 2015 - A generalized back-door criterion.pdf:application/pdf},
}

@inproceedings{entner_data-driven_2013,
	title = {Data-driven covariate selection for nonparametric estimation of causal eﬀects},
	abstract = {The estimation of causal eﬀects from nonexperimental data is a fundamental problem in many ﬁelds of science. One of the main obstacles concerns confounding by observed or latent covariates, an issue which is typically tackled by adjusting for some set of observed covariates. In this contribution, we analyze the problem of inferring whether a given variable has a causal eﬀect on another and, if it does, inferring an adjustment set of covariates that yields a consistent and unbiased estimator of this eﬀect, based on the (conditional) independence and dependence relationships among the observed variables. We provide two elementary rules that we show to be both sound and complete for this task, and compare the performance of a straightforward application of these rules with standard alternative procedures for selecting adjustment sets.},
	language = {en},
	booktitle = {Proceedings of the 16th {International} {Conference} on {Artificial} {Intelligence} and {Statistics} ({AISTATS})},
	author = {Entner, Doris and Hoyer, Patrik O and Spirtes, Peter},
	year = {2013},
	file = {Entner et al. - Data-driven covariate selection for nonparametric .pdf:/Users/jacquelinemaasch/Zotero/storage/5KPDY27R/Entner et al. - Data-driven covariate selection for nonparametric .pdf:application/pdf},
}

@misc{pearl_class_2012,
	title = {On a {Class} of {Bias}-{Amplifying} {Variables} that {Endanger} {Eﬀect} {Estimates}},
	url = {https://arxiv-org.proxy.library.cornell.edu/pdf/1203.3503.pdf},
	language = {en},
	author = {Pearl, Judea},
	year = {2012},
	file = {Pearl - On a Class of Bias-Amplifying Variables that Endan.pdf:/Users/jacquelinemaasch/Zotero/storage/RCKZFNH3/Pearl - On a Class of Bias-Amplifying Variables that Endan.pdf:application/pdf},
}

@article{cheng_estimating_2021,
	title = {Estimating the natural indirect effect and the mediation proportion via the product method},
	volume = {21},
	issn = {1471-2288},
	url = {https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-021-01425-4},
	doi = {10.1186/s12874-021-01425-4},
	abstract = {Background: The natural indirect effect (NIE) and mediation proportion (MP) are two measures of primary interest in mediation analysis. The standard approach for mediation analysis is through the product method, which involves a model for the outcome conditional on the mediator and exposure and another model describing the exposure–mediator relationship. The purpose of this article is to comprehensively develop and investigate the finite-sample performance of NIE and MP estimators via the product method.
Methods: With four common data types with a continuous/binary outcome and a continuous/binary mediator, we propose closed-form interval estimators for NIE and MP via the theory of multivariate delta method, and evaluate its empirical performance relative to the bootstrap approach. In addition, we have observed that the rare outcome assumption is frequently invoked to approximate the NIE and MP with a binary outcome, although this approximation may lead to non-negligible bias when the outcome is common. We therefore introduce the exact expressions for NIE and MP with a binary outcome without the rare outcome assumption and compare its performance with the approximate estimators.
Results: Simulation studies suggest that the proposed interval estimator provides satisfactory coverage when the sample size ≥500 for the scenarios with a continuous outcome and sample size ≥20,000 and number of cases ≥500 for the scenarios with a binary outcome. In the binary outcome scenarios, the approximate estimators based on the rare outcome assumption worked well when outcome prevalence less than 5\% but could lead to substantial bias when the outcome is common; in contrast, the exact estimators always perform well under all outcome prevalences considered.
Conclusions: Under samples sizes commonly encountered in epidemiology and public health research, the proposed interval estimator is valid for constructing confidence interval. For a binary outcome, the exact estimator without the rare outcome assumption is more robust and stable to estimate NIE and MP. An R package mediateP is developed to implement the methods for point and variance estimation discussed in this paper.},
	language = {en},
	number = {1},
	urldate = {2023-04-26},
	journal = {BMC Medical Research Methodology},
	author = {Cheng, Chao and Spiegelman, Donna and Li, Fan},
	month = nov,
	year = {2021},
	pages = {253},
	file = {Cheng et al. - 2021 - Estimating the natural indirect effect and the med.pdf:/Users/jacquelinemaasch/Zotero/storage/I7FXJ7G6/Cheng et al. - 2021 - Estimating the natural indirect effect and the med.pdf:application/pdf},
}

@inproceedings{pearl_direct_2001,
	title = {Direct and {Indirect} {Effects}},
	booktitle = {Proceedings of the {Seventeenth} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Pearl, Judea},
	year = {2001},
	file = {Pearl - 2001 - Direct and Indirect Effects.pdf:/Users/jacquelinemaasch/Zotero/storage/ZTJHIUHM/Pearl - 2001 - Direct and Indirect Effects.pdf:application/pdf},
}

@article{hernan_structural_2004,
	title = {A {Structural} {Approach} to {Selection} {Bias}},
	volume = {15},
	issn = {1044-3983},
	shorttitle = {A {Structural} {Approach} to {Selection} {Bias}},
	url = {http://journals.lww.com/00001648-200409000-00020},
	doi = {10.1097/01.ede.0000135174.63482.43},
	abstract = {The term “selection bias” encompasses various biases in epidemiology. We describe examples of selection bias in casecontrol studies (eg, inappropriate selection of controls) and cohort studies (eg, informative censoring). We argue that the causal structure underlying the bias in each example is essentially the same: conditioning on a common effect of 2 variables, one of which is either exposure or a cause of exposure and the other is either the outcome or a cause of the outcome. This structure is shared by other biases (eg, adjustment for variables affected by prior exposure). A structural classiﬁcation of bias distinguishes between biases resulting from conditioning on common effects (“selection bias”) and those resulting from the existence of common causes of exposure and outcome (“confounding”). This classiﬁcation also leads to a uniﬁed approach to adjust for selection bias.},
	language = {en},
	number = {5},
	urldate = {2023-04-26},
	journal = {Epidemiology},
	author = {Hernán, Miguel A. and Hernández-Díaz, Sonia and Robins, James M.},
	month = sep,
	year = {2004},
	pages = {615--625},
	file = {Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:/Users/jacquelinemaasch/Zotero/storage/FVCALRE7/Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:application/pdf;Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:/Users/jacquelinemaasch/Zotero/storage/BH49ZQLC/Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:application/pdf;Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:/Users/jacquelinemaasch/Zotero/storage/ZHYR8DQN/Hernán et al. - 2004 - A Structural Approach to Selection Bias.pdf:application/pdf},
}

@article{greenland_confounding_2001,
	title = {Confounding in {Health} {Research}},
	volume = {22},
	issn = {0163-7525, 1545-2093},
	url = {https://www.annualreviews.org/doi/10.1146/annurev.publhealth.22.1.189},
	doi = {10.1146/annurev.publhealth.22.1.189},
	abstract = {▪ Abstract  Consideration of confounding is fundamental to the design, analysis, and interpretation of studies intended to estimate causal effects. Unfortunately, the word confounding has been used synonymously with several other terms, and it has been used to refer to at least four distinct concepts. This paper provides an overview of confounding and related concepts based on a counterfactual model of causation. In this context, which predominates in nonexperimental research, confounding is a source of bias in the estimation of causal effects. Special attention is given to the history of definitions of confounding, the distinction between confounding and confounders, problems in the control of confounding, the relations of confounding to exchangeability and collapsibility, and confounding in randomized trials.},
	language = {en},
	number = {1},
	urldate = {2023-04-26},
	journal = {Annual Review of Public Health},
	author = {Greenland, Sander and Morgenstern, Hal},
	month = may,
	year = {2001},
	pages = {189--212},
	file = {Greenland and Morgenstern - 2001 - Confounding in Health Research.pdf:/Users/jacquelinemaasch/Zotero/storage/9NE7T86B/Greenland and Morgenstern - 2001 - Confounding in Health Research.pdf:application/pdf},
}

@article{pearl_causal_1995,
	title = {Causal diagrams for empirical research},
	volume = {82},
	language = {en},
	number = {4},
	journal = {Biometrika},
	author = {Pearl, Judea},
	year = {1995},
	pages = {669--688},
	file = {Pearl - Causal diagrams for empirical research.pdf:/Users/jacquelinemaasch/Zotero/storage/FP5LCRJ3/Pearl - Causal diagrams for empirical research.pdf:application/pdf},
}

@article{rubin_for_2008,
	title = {For objective causal inference, design trumps analysis},
	volume = {2},
	issn = {1932-6157},
	url = {https://projecteuclid.org/journals/annals-of-applied-statistics/volume-2/issue-3/For-objective-causal-inference-design-trumps-analysis/10.1214/08-AOAS187.full},
	doi = {10.1214/08-AOAS187},
	language = {en},
	number = {3},
	urldate = {2023-04-26},
	journal = {The Annals of Applied Statistics},
	author = {Rubin, Donald B.},
	month = sep,
	year = {2008},
	file = {Rubin - 2008 - For objective causal inference, design trumps anal.pdf:/Users/jacquelinemaasch/Zotero/storage/X76MNQYC/Rubin - 2008 - For objective causal inference, design trumps anal.pdf:application/pdf},
}

@article{liu_implications_2012,
	title = {Implications of {M} {Bias} in {Epidemiologic} {Studies}: {A} {Simulation} {Study}},
	volume = {176},
	issn = {1476-6256, 0002-9262},
	shorttitle = {Implications of {M} {Bias} in {Epidemiologic} {Studies}},
	url = {https://academic.oup.com/aje/article/176/10/938/92975},
	doi = {10.1093/aje/kws165},
	abstract = {Abstract
            Collider-stratification bias arises from conditioning on a variable (collider) which opens a path from exposure to outcome. M bias occurs when the collider-stratification bias is transmitted through ancestors of exposure and outcome. Previous theoretical work, but not empirical data, has demonstrated that M bias is smaller than confounding bias. The authors simulated data for large cohort studies with binary exposure, an outcome, a collider, and 2 predictors of the collider. They created 178 scenarios by changing the frequencies of variables and/or the magnitudes of associations among the variables. They calculated the effect estimate, percentage bias, and mean squared error. M bias in these realistic scenarios ranged from −2\% to −5\%. When the authors increased one or both relative risks for the relation between the collider and unmeasured factors to ≥8, the negative bias was more substantial (\&gt;15\%). The result was substantially biased (e.g., \&gt;20\%) if an unmeasured confounder that was also a collider was not adjusted to avoid M bias. In scenarios resembling those the authors examined, M bias had a small impact unless associations between the collider and unmeasured confounders were very large (relative risk \&gt; 8). When a collider is itself an important confounder, controlling for confounding would take precedence over avoiding M bias.},
	language = {en},
	number = {10},
	urldate = {2023-04-26},
	journal = {American Journal of Epidemiology},
	author = {Liu, Wei and Brookhart, M. Alan and Schneeweiss, Sebastian and Mi, Xiaojuan and Setoguchi, Soko},
	month = nov,
	year = {2012},
	pages = {938--948},
	file = {Liu et al. - 2012 - Implications of M Bias in Epidemiologic Studies A.pdf:/Users/jacquelinemaasch/Zotero/storage/9NHD66N5/Liu et al. - 2012 - Implications of M Bias in Epidemiologic Studies A.pdf:application/pdf;Liu et al. - 2012 - Implications of M Bias in Epidemiologic Studies A.pdf:/Users/jacquelinemaasch/Zotero/storage/GMA3EFTA/Liu et al. - 2012 - Implications of M Bias in Epidemiologic Studies A.pdf:application/pdf},
}

@article{greenland_introduction_2000,
	title = {An introduction to instrumental variables for epidemiologists},
	volume = {29},
	language = {en},
	journal = {International Journal of Epidemiology},
	author = {Greenland, Sander},
	year = {2000},
	pages = {722--729},
	file = {Greenland - An introduction to instrumental variables for epid.pdf:/Users/jacquelinemaasch/Zotero/storage/H3B8GG3R/Greenland - An introduction to instrumental variables for epid.pdf:application/pdf},
}

@article{meyer_information-theoretic_2008,
	title = {Information-{Theoretic} {Feature} {Selection} in {Microarray} {Data} {Using} {Variable} {Complementarity}},
	volume = {2},
	issn = {1932-4553},
	url = {http://ieeexplore.ieee.org/document/4550559/},
	doi = {10.1109/JSTSP.2008.923858},
	abstract = {The paper presents an original ﬁlter approach for effective feature selection in microarray data characterized by a large number of input variables and a few samples. The approach is based on the use of a new information-theoretic selection, the Double Input Symmetrical Relevance (DISR), which relies on a measure of variable complementarity. This measure evaluates the additional information that a set of variables provides about the output with respect to the sum of each single variable contribution. We show that a variable selection approach based on DISR can be formulated as a quadratic optimization problem: the Dispersion Sum Problem (DSP). To solve this problem, we use a strategy based on Backward Elimination and Sequential Replacement (BESR). The combination of BESR and the DISR criterion is compared in theoretical and experimental terms to recently proposed information-theoretic criteria. Experimental results on a synthetic dataset as well as on a set of eleven microarray classiﬁcation tasks show that the proposed technique is competitive with existing ﬁlter selection methods.},
	language = {en},
	number = {3},
	urldate = {2023-04-27},
	journal = {IEEE Journal of Selected Topics in Signal Processing},
	author = {Meyer, Patrick Emmanuel and Schretter, Colas and Bontempi, Gianluca},
	month = jun,
	year = {2008},
	pages = {261--274},
	file = {Meyer et al. - 2008 - Information-Theoretic Feature Selection in Microar.pdf:/Users/jacquelinemaasch/Zotero/storage/JWXZGSGP/Meyer et al. - 2008 - Information-Theoretic Feature Selection in Microar.pdf:application/pdf},
}

@book{frieze_introduction_2015,
	edition = {1},
	title = {Introduction to {Random} {Graphs}},
	isbn = {978-1-107-11850-8 978-1-316-33983-1},
	url = {https://www.cambridge.org/core/product/identifier/9781316339831/type/book},
	abstract = {From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.},
	urldate = {2023-05-02},
	publisher = {Cambridge University Press},
	author = {Frieze, Alan and Karoński, Michał},
	month = oct,
	year = {2015},
	doi = {10.1017/CBO9781316339831},
	file = {Frieze and Karoński - 2015 - Introduction to Random Graphs.pdf:/Users/jacquelinemaasch/Zotero/storage/K8ITUWN5/Frieze and Karoński - 2015 - Introduction to Random Graphs.pdf:application/pdf},
}

@article{soderberg_general_2002,
	title = {A {General} {Formalism} for {Inhomogeneous} {Random} {Graphs}},
	volume = {66},
	issn = {1063-651X, 1095-3787},
	url = {http://arxiv.org/abs/cond-mat/0211063},
	doi = {10.1103/PhysRevE.66.066121},
	abstract = {We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its terminal vertices. This approach provides a general framework for the analysis of a large class of models. The generic phase structure is derived using generating function techniques, and relations to other classes of models are pointed out.},
	language = {en},
	number = {6},
	urldate = {2023-05-02},
	journal = {Physical Review E},
	author = {Soderberg, Bo},
	month = dec,
	year = {2002},
	note = {arXiv:cond-mat/0211063},
	keywords = {Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Statistical Mechanics},
	pages = {066121},
	file = {Soderberg - 2002 - A General Formalism for Inhomogeneous Random Graph.pdf:/Users/jacquelinemaasch/Zotero/storage/VWUTAPG5/Soderberg - 2002 - A General Formalism for Inhomogeneous Random Graph.pdf:application/pdf},
}

@misc{sigtermans_is_2020,
	title = {Is {Information} {Theory} {Inherently} a {Theory} of {Causation}?},
	url = {http://arxiv.org/abs/2010.01932},
	abstract = {Information theory gives rise to a novel method for causal skeleton discovery by expressing associations between variables as tensors. This tensor-based approach reduces the dimensionality of the data needed to test for conditional independence, e.g., for systems comprising three variables, the causal skeleton can be determined using pair-wise determined tensors. To arrive at this result, an additional information measure, path information, is proposed.},
	language = {en},
	urldate = {2023-05-10},
	publisher = {arXiv},
	author = {Sigtermans, David},
	month = nov,
	year = {2020},
	note = {arXiv:2010.01932 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Information Theory},
	file = {Sigtermans - 2020 - Is Information Theory Inherently a Theory of Causa.pdf:/Users/jacquelinemaasch/Zotero/storage/HKT4PHNX/Sigtermans - 2020 - Is Information Theory Inherently a Theory of Causa.pdf:application/pdf},
}

@article{sigtermans_determining_2020,
	title = {Determining {Causal} {Skeletons} with {Information} {Theory}},
	volume = {23},
	issn = {1099-4300},
	url = {https://www.mdpi.com/1099-4300/23/1/38},
	doi = {10.3390/e23010038},
	abstract = {Modeling a causal association as arising from a communication process between cause and effect, simpliﬁes the discovery of causal skeletons. The communication channels enabling these communication processes, are fully characterized by stochastic tensors, and therefore allow us to use linear algebra. This tensor-based approach reduces the dimensionality of the data needed to test for conditional independence, e.g., for systems comprising three variables, pair-wise determined tensors sufﬁce to infer the causal skeleton. The only thing needed is a minor extension to information theory, namely the concept of path information.},
	language = {en},
	number = {1},
	urldate = {2023-05-10},
	journal = {Entropy},
	author = {Sigtermans, David},
	month = dec,
	year = {2020},
	pages = {38},
	file = {Sigtermans - 2020 - Determining Causal Skeletons with Information Theo.pdf:/Users/jacquelinemaasch/Zotero/storage/4TET5C6U/Sigtermans - 2020 - Determining Causal Skeletons with Information Theo.pdf:application/pdf},
}

@article{paninski_estimation_2003,
	title = {Estimation of {Entropy} and {Mutual} {Information}},
	volume = {15},
	issn = {0899-7667, 1530-888X},
	url = {https://direct.mit.edu/neco/article/15/6/1191-1253/6731},
	doi = {10.1162/089976603321780272},
	abstract = {We present some new results on the nonparametric estimation of entropy and mutual information. First, we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators. The setup is related to Grenander's method of sieves and places no assumptions on the underlying probability measure generating the data. Second, we prove a converse to these consistency theorems, demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information, even given unlimited data. This “inconsistency” theorem leads to an analytical approximation of the bias, valid in surprisingly small sample regimes and more accurate than the usual [Formula: see text] formula of Miller and Madow over a large region of parameter space. The two most practical implications of these results are negative: (1) information estimates in a certain data regime are likely contaminated by bias, even if “bias-corrected” estimators are used, and (2) confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods.
            Finally, we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem. By casting bias calculation problems in this approximation theory framework, we obtain the best possible generalization of known asymptotic bias results. More interesting, this framework leads to an estimator with some nice properties: the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions, and this maximum error turns out to be surprisingly small. We demonstrate the application of this new estimator on both real and simulated data.},
	language = {en},
	number = {6},
	urldate = {2023-05-10},
	journal = {Neural Computation},
	author = {Paninski, Liam},
	month = jun,
	year = {2003},
	pages = {1191--1253},
	file = {Paninski - 2003 - Estimation of Entropy and Mutual Information.pdf:/Users/jacquelinemaasch/Zotero/storage/SV882NV8/Paninski - 2003 - Estimation of Entropy and Mutual Information.pdf:application/pdf},
}

@article{compton_information-theoretic_nodate,
	title = {Information-{Theoretic} {Algorithms} and {Identifiability} for {Causal} {Graph} {Discovery}},
	abstract = {It is a task of widespread interest to learn the underlying causal structure for systems of random variables. Entropic Causal Inference is a recent framework for learning the causal graph between two variables from observational data (i.e., without experiments) by finding the information-theoretically simplest structural explanation of the data. In this thesis, we develop theoretical techniques that enable us to show how Entropic Causal Inference permits learnability of causal graphs with particular informationtheoretically simple structure. We show the first theoretical guarantee for finite-sample learnability with Entropic Causal Inference for pairs of random variables. Later, we extend this guarantee to show the first result for Entropic Causal Inference in systems with more than two variables: proving learnability of general directed acyclic graphs over many variables (under assumptions on the generative process). We implement and experimentally evaluate Entropic Causal Inference on synthetic and real-world causal systems. Moreover, we improve the best-known approximation guarantee for the Minimum Entropy Coupling problem. This information-theoretic algorithmic problem has direct relevance to Entropic Causal Inference and is also of independent interest. In totality, this thesis develops algorithmic and information-theoretic tools that shed light on how information-theoretic properties enable learning of causal graphs from both a practical and theoretical perspective.},
	language = {en},
	author = {Compton, Spencer},
	file = {Compton - Information-Theoretic Algorithms and Identifiabili.pdf:/Users/jacquelinemaasch/Zotero/storage/MIDBLJSI/Compton - Information-Theoretic Algorithms and Identifiabili.pdf:application/pdf},
}

@article{janzing_causal_2010,
	title = {Causal {Inference} {Using} the {Algorithmic} {Markov} {Condition}},
	volume = {56},
	issn = {0018-9448, 1557-9654},
	url = {http://ieeexplore.ieee.org/document/5571886/},
	doi = {10.1109/TIT.2010.2060095},
	abstract = {Inferring the causal structure that links observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why causal inference is also possible when the sample size is one. We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules. We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work. We also sketch some ideas on how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution.},
	language = {en},
	number = {10},
	urldate = {2023-05-10},
	journal = {IEEE Transactions on Information Theory},
	author = {Janzing, Dominik and Scholkopf, Bernhard},
	month = oct,
	year = {2010},
	pages = {5168--5194},
	file = {Janzing and Scholkopf - 2010 - Causal Inference Using the Algorithmic Markov Cond.pdf:/Users/jacquelinemaasch/Zotero/storage/E3U22EH3/Janzing and Scholkopf - 2010 - Causal Inference Using the Algorithmic Markov Cond.pdf:application/pdf},
}

@article{mackay_information_nodate,
	title = {Information {Theory}, {Inference}, and {Learning} {Algorithms}},
	language = {en},
	author = {MacKay, David J C},
	file = {MacKay - Information Theory, Inference, and Learning Algori.pdf:/Users/jacquelinemaasch/Zotero/storage/7CP6YFH6/MacKay - Information Theory, Inference, and Learning Algori.pdf:application/pdf},
}

@book{hayes_introduction_2018,
	address = {New York},
	edition = {Second edition},
	series = {Methodology in the social sciences},
	title = {Introduction to mediation, moderation, and conditional process analysis: a regression-based approach},
	isbn = {978-1-4625-3465-4},
	shorttitle = {Introduction to mediation, moderation, and conditional process analysis},
	language = {en},
	publisher = {Guilford Press},
	author = {Hayes, Andrew F.},
	year = {2018},
	keywords = {Mediation (Statistics), Regression analysis, Social sciences, Statistical methods},
	file = {Hayes - 2018 - Introduction to mediation, moderation, and conditi.pdf:/Users/jacquelinemaasch/Zotero/storage/68KBP7SV/Hayes - 2018 - Introduction to mediation, moderation, and conditi.pdf:application/pdf;Hayes - 2018 - Introduction to mediation, moderation, and conditi.pdf:/Users/jacquelinemaasch/Zotero/storage/PI89T7BY/Hayes - 2018 - Introduction to mediation, moderation, and conditi.pdf:application/pdf},
}

@misc{purkayastha_asymmetric_2022,
	title = {Asymmetric predictability in causal discovery: an information theoretic approach},
	shorttitle = {Asymmetric predictability in causal discovery},
	url = {http://arxiv.org/abs/2210.14455},
	abstract = {Causal investigations in observational studies pose a great challenge in scientiﬁc research where randomized trials or intervention-based studies are not feasible. Leveraging Shannon’s seminal work on information theory, we develop a causal discovery framework of “predictive asymmetry” for bivariate (X, Y ). Predictive asymmetry is a central concept in information geometric causal inference; it enables assessment of whether X is a stronger predictor of Y or vice-versa. We propose a new metric called the Asymmetric Mutual Information (AM I) and establish its key statistical properties. The AM I is not only able to detect complex non-linear association patterns in bivariate data, but also is able to detect and quantify predictive asymmetry. Our proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation. The resulting estimation method is manyfold faster than the classical bandwidth-based density estimation, while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of the AM I methodology; a new data-splitting technique is developed to make statistical inference on predictive asymmetry using the AM I. We illustrate the performance of the AM I methodology through simulation studies as well as multiple real data examples.},
	language = {en},
	urldate = {2023-05-11},
	publisher = {arXiv},
	author = {Purkayastha, Soumik and Song, Peter X. K.},
	month = oct,
	year = {2022},
	note = {arXiv:2210.14455 [math, stat]},
	keywords = {Statistics - Methodology, Mathematics - Statistics Theory, Statistics - Computation},
	file = {Purkayastha and Song - 2022 - Asymmetric predictability in causal discovery an .pdf:/Users/jacquelinemaasch/Zotero/storage/CBTDZTL4/Purkayastha and Song - 2022 - Asymmetric predictability in causal discovery an .pdf:application/pdf},
}

@misc{cheng_towards_2020,
	title = {Towards unique and unbiased causal effect estimation from data with hidden variables},
	url = {http://arxiv.org/abs/2002.10091},
	abstract = {Causal effect estimation from observational data is a crucial but challenging task. Currently, only a limited number of data-driven causal effect estimation methods are available. These methods either provide only a bound estimation of the causal effect of a treatment on the outcome, or generate a unique estimation of the causal effect, but making strong assumptions on data and having low efﬁciency. In this paper, we identify a practical problem setting and propose an approach to achieving unique and unbiased estimation of causal effects from data with hidden variables. For the approach, we have developed the theorems to support the discovery of the proper covariate sets for confounding adjustment (adjustment sets). Based on the theorems, two algorithms are proposed for ﬁnding the proper adjustment sets from data with hidden variables to obtain unbiased and unique causal effect estimation. Experiments with synthetic datasets generated using ﬁve benchmark Bayesian networks and four real-world datasets have demonstrated the efﬁciency and effectiveness of the proposed algorithms, indicating the practicability of the identiﬁed problem setting and the potential of the proposed approach in real-world applications.},
	language = {en},
	urldate = {2023-05-11},
	publisher = {arXiv},
	author = {Cheng, Debo and Li, Jiuyong and Liu, Lin and Yu, Kui and Lee, Thuc Duy and Liu, Jixue},
	month = nov,
	year = {2020},
	note = {arXiv:2002.10091 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Statistics - Methodology},
	file = {Cheng et al. - 2020 - Towards unique and unbiased causal effect estimati.pdf:/Users/jacquelinemaasch/Zotero/storage/PVE68P39/Cheng et al. - 2020 - Towards unique and unbiased causal effect estimati.pdf:application/pdf},
}

@misc{shah_finding_2022,
	title = {Finding {Valid} {Adjustments} under {Non}-ignorability with {Minimal} {DAG} {Knowledge}},
	url = {http://arxiv.org/abs/2106.11560},
	abstract = {Treatment eﬀect estimation from observational data is a fundamental problem in causal inference. There are two very diﬀerent schools of thought that have tackled this problem. On the one hand, the Pearlian framework commonly assumes structural knowledge (provided by an expert) in the form of directed acyclic graphs and provides graphical criteria such as the back-door criterion to identify the valid adjustment sets. On the other hand, the potential outcomes (PO) framework commonly assumes that all the observed features satisfy ignorability (i.e., no hidden confounding), which in general is untestable. In prior works that attempted to bridge these frameworks, there is an observational criteria to identify an anchor variable and if a subset of covariates (not involving the anchor variable) passes a suitable conditional independence criteria, then that subset is a valid back-door. Our main result strengthens these prior results by showing that under a diﬀerent expert-driven structural knowledge — that one variable is a direct causal parent of the treatment variable — remarkably, testing for subsets (not involving the known parent variable) that are valid back-doors is equivalent to an invariance test. Importantly, we also cover the non-trivial case where the entire set of observed features is not ignorable (generalizing the PO framework) without requiring the knowledge of all the parents of the treatment variable. Our key technical idea involves generation of a synthetic sub-sampling (or environment) variable that is a function of the known parent variable. In addition to designing an invariance test, this sub-sampling variable allows us to leverage Invariant Risk Minimization, and thus, connects ﬁnding valid adjustments (in non-ignorable observational settings) to representation learning. We demonstrate the eﬀectiveness and tradeoﬀs of these approaches on a variety of synthetic datasets as well as real causal eﬀect estimation benchmarks.},
	language = {en},
	urldate = {2023-05-11},
	publisher = {arXiv},
	author = {Shah, Abhin and Shanmugam, Karthikeyan and Ahuja, Kartik},
	month = feb,
	year = {2022},
	note = {arXiv:2106.11560 [cs]},
	keywords = {Computer Science - Machine Learning},
	file = {Shah et al. - 2022 - Finding Valid Adjustments under Non-ignorability w.pdf:/Users/jacquelinemaasch/Zotero/storage/FXYR6JQA/Shah et al. - 2022 - Finding Valid Adjustments under Non-ignorability w.pdf:application/pdf},
}

@article{henckel_graphical_2022,
	title = {Graphical {Criteria} for {Efficient} {Total} {Effect} {Estimation} via {Adjustment} in {Causal} {Linear} {Models}},
	volume = {84},
	issn = {1369-7412, 1467-9868},
	url = {http://arxiv.org/abs/1907.02435},
	doi = {10.1111/rssb.12451},
	abstract = {Covariate adjustment is a commonly used method for total causal eﬀect estimation. In recent years, graphical criteria have been developed to identify all valid adjustment sets, that is, all covariate sets that can be used for this purpose. Diﬀerent valid adjustment sets typically provide total eﬀect estimates of varying accuracies. Restricting ourselves to causal linear models, we introduce a graphical criterion to compare the asymptotic variances provided by certain valid adjustment sets. We employ this result to develop two further graphical tools. First, we introduce a simple variance reducing pruning procedure for any given valid adjustment set. Second, we give a graphical characterization of a valid adjustment set that provides the optimal asymptotic variance among all valid adjustment sets. Our results depend only on the graphical structure and not on the speciﬁc error variances or edge coeﬃcients of the underlying causal linear model. They can be applied to directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs) and maximally oriented partially directed acyclic graphs (maximal PDAGs). We present simulations and a real data example to support our results and show their practical applicability.},
	language = {en},
	number = {2},
	urldate = {2023-05-11},
	journal = {Journal of the Royal Statistical Society Series B: Statistical Methodology},
	author = {Henckel, Leonard and Perković, Emilija and Maathuis, Marloes H.},
	month = apr,
	year = {2022},
	note = {arXiv:1907.02435 [math, stat]},
	keywords = {Mathematics - Statistics Theory},
	pages = {579--599},
	file = {Henckel et al. - 2022 - Graphical Criteria for Efficient Total Effect Esti.pdf:/Users/jacquelinemaasch/Zotero/storage/3I4B9TL2/Henckel et al. - 2022 - Graphical Criteria for Efficient Total Effect Esti.pdf:application/pdf},
}

@misc{rotnitzky_efficient_2019,
	title = {Efficient adjustment sets for population average treatment effect estimation in non-parametric causal graphical models},
	url = {http://arxiv.org/abs/1912.00306},
	abstract = {The method of covariate adjustment is often used for estimation of population average causal treatment eﬀects in observational studies. Graphical rules for determining all valid covariate adjustment sets from an assumed causal graphical model are well known. Restricting attention to causal linear models, a recent article (Henckel et al., 2019) derived two novel graphical criteria: one to compare the asymptotic variance of linear regression treatment eﬀect estimators that control for certain distinct adjustment sets and another to identify the optimal adjustment set that yields the adjusted least squares treatment eﬀect estimator with the smallest asymptotic variance among consistent adjusted least squares estimators. In this paper we show that the same graphical criteria can be used in non-parametric causal graphical models when treatment eﬀects are estimated by contrasts involving non-parametrically adjusted estimators of the interventional means. We also provide a graphical criterion for determining the optimal adjustment set among the minimal adjustment sets, which is valid for both linear and non-parametric estimators. We additionally provide a new graphical criterion for comparing time dependent adjustment sets, that is, sets comprised by covariates that adjust for future treatments and that are themselves aﬀected by earlier treatments. We show by example that uniformly optimal time dependent adjustment sets do not always exist. In addition, for point interventions, we provide a sound and complete graphical criterion for determining when a non-parametric optimally adjusted estimator of an interventional mean, or of a contrast of interventional means, is as eﬃcient as an eﬃcient estimator of the same parameter that exploits the information in the conditional independencies encoded in the non-parametric causal graphical model. The algorithm also checks for possible simpliﬁcations of the eﬃcient inﬂuence function of the parameter. We ﬁnd an interesting connection between identiﬁcation and eﬃcient covariate adjustment estimation. Speciﬁcally, we show that if there exists an identifying formula for an interventional mean that depends only on treatment, outcome and mediators, then the non-parametric optimally adjusted estimator can never be globally eﬃcient under the non-parametric causal graphical model.},
	language = {en},
	urldate = {2023-05-11},
	publisher = {arXiv},
	author = {Rotnitzky, Andrea and Smucler, Ezequiel},
	month = dec,
	year = {2019},
	note = {arXiv:1912.00306 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Mathematics - Statistics Theory},
	file = {Rotnitzky and Smucler - 2019 - Efficient adjustment sets for population average t.pdf:/Users/jacquelinemaasch/Zotero/storage/8UMULV7T/Rotnitzky and Smucler - 2019 - Efficient adjustment sets for population average t.pdf:application/pdf},
}

@misc{gupta_estimating_2021,
	title = {Estimating {Treatment} {Effects} with {Observed} {Confounders} and {Mediators}},
	url = {http://arxiv.org/abs/2003.11991},
	abstract = {Given a causal graph, the do-calculus can express treatment effects as functionals of the observational joint distribution that can be estimated empirically. Sometimes the do-calculus identiﬁes multiple valid formulae, prompting us to compare the statistical properties of the corresponding estimators. For example, the backdoor formula applies when all confounders are observed and the frontdoor formula applies when an observed mediator transmits the causal effect. In this paper, we investigate the over-identiﬁed scenario where both confounders and mediators are observed, rendering both estimators valid. Addressing the linear Gaussian causal model, we demonstrate that either estimator can dominate the other by an unbounded constant factor. Next, we derive an optimal estimator, which leverages all observed variables, and bound its ﬁnite-sample variance. We show that it strictly outperforms the backdoor and frontdoor estimators and that this improvement can be unbounded. We also present a procedure for combining two datasets, one with observed confounders and another with observed mediators. Finally, we evaluate our methods on both simulated data and the IHDP and JTPA datasets.},
	language = {en},
	urldate = {2023-05-11},
	publisher = {arXiv},
	author = {Gupta, Shantanu and Lipton, Zachary C. and Childers, David},
	month = jun,
	year = {2021},
	note = {arXiv:2003.11991 [cs, econ, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology, Economics - Econometrics},
	file = {Gupta et al. - 2021 - Estimating Treatment Effects with Observed Confoun.pdf:/Users/jacquelinemaasch/Zotero/storage/ZWUAVUP9/Gupta et al. - 2021 - Estimating Treatment Effects with Observed Confoun.pdf:application/pdf},
}

@inproceedings{gupta_local_2023,
	title = {Local {Causal} {Discovery} for {Estimating} {Causal} {Effects}},
	url = {http://arxiv.org/abs/2302.08070},
	abstract = {Even when the causal graph underlying our data is unknown, we can use observational data to narrow down the possible values that an average treatment effect (ATE) can take by (1) identifying the graph up to a Markov equivalence class; and (2) estimating that ATE for each graph in the class. While the PC algorithm can identify this class under strong faithfulness assumptions, it can be computationally prohibitive. Fortunately, only the local graph structure around the treatment is required to identify the set of possible ATE values, a fact exploited by local discovery algorithms to improve computational efﬁciency. In this paper, we introduce Local Discovery using Eager Collider Checks (LDECC), a new local causal discovery algorithm that leverages unshielded colliders to orient the treatment’s parents differently from existing methods. We show that there exist graphs where LDECC exponentially outperforms existing local discovery algorithms and vice versa. Moreover, we show that LDECC and existing algorithms rely on different faithfulness assumptions, leveraging this insight to weaken the assumptions for identifying the set of possible ATE values.},
	language = {en},
	urldate = {2023-05-11},
	booktitle = {Proceedings of the 2nd {Conference} on {Causal} {Learning} and {Reasoning} ({CLeaR})},
	publisher = {arXiv},
	author = {Gupta, Shantanu and Childers, David and Lipton, Zachary C.},
	month = feb,
	year = {2023},
	note = {arXiv:2302.08070 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Gupta et al. - 2023 - Local Causal Discovery for Estimating Causal Effec.pdf:/Users/jacquelinemaasch/Zotero/storage/RQEEJJHM/Gupta et al. - 2023 - Local Causal Discovery for Estimating Causal Effec.pdf:application/pdf},
}

@article{runge_necessary_2021,
	title = {Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables},
	abstract = {The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables. For the case with hidden variables there can be settings where no optimal set exists and currently only a sufficient graphical optimality criterion of limited applicability has been derived. In the present work optimality is characterized as maximizing a certain adjustment information which allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set and a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has higher (or equal) adjustment information than the Adjust-set proposed in Perkovic´ et al. [Journal of Machine Learning Research, 18: 1–62, 2018] for any graph. The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation. Numerical experiments indicate that the asymptotic results also hold for relatively small sample sizes and that the optimal adjustment set or minimized variants thereof often yield better variance also beyond that estimator class. Surprisingly, among the randomly created setups more than 90\% fulfill the optimality conditions indicating that also in many real-world scenarios graphical optimality may hold.},
	language = {en},
	journal = {Proceedings of the 35th Conference on Neural Information Processing Systems},
	author = {Runge, Jakob},
	year = {2021},
	file = {Runge - Necessary and sufficient graphical conditions for .pdf:/Users/jacquelinemaasch/Zotero/storage/H9ITB38T/Runge - Necessary and sufficient graphical conditions for .pdf:application/pdf},
}

@article{howards_toward_2012,
	title = {"{Toward} a {Clearer} {Definition} of {Confounding}" {Revisited} {With} {Directed} {Acyclic} {Graphs}},
	volume = {176},
	issn = {0002-9262, 1476-6256},
	url = {https://academic.oup.com/aje/article-lookup/doi/10.1093/aje/kws127},
	doi = {10.1093/aje/kws127},
	language = {en},
	number = {6},
	urldate = {2023-05-11},
	journal = {American Journal of Epidemiology},
	author = {Howards, P. P. and Schisterman, E. F. and Poole, C. and Kaufman, J. S. and Weinberg, C. R.},
	month = sep,
	year = {2012},
	pages = {506--511},
	file = {Howards et al. - 2012 - Toward a Clearer Definition of Confounding Revis.pdf:/Users/jacquelinemaasch/Zotero/storage/F35PYRP9/Howards et al. - 2012 - Toward a Clearer Definition of Confounding Revis.pdf:application/pdf},
}

@article{schreiber_measuring_2000,
	title = {Measuring {Information} {Transfer}},
	volume = {85},
	issn = {0031-9007, 1079-7114},
	url = {https://link.aps.org/doi/10.1103/PhysRevLett.85.461},
	doi = {10.1103/PhysRevLett.85.461},
	language = {en},
	number = {2},
	urldate = {2023-05-11},
	journal = {Physical Review Letters},
	author = {Schreiber, Thomas},
	month = jul,
	year = {2000},
	pages = {461--464},
	file = {Schreiber - 2000 - Measuring Information Transfer.pdf:/Users/jacquelinemaasch/Zotero/storage/GFDCA84W/Schreiber - 2000 - Measuring Information Transfer.pdf:application/pdf},
}

@article{marko_bidirectional_1973,
	title = {The {Bidirectional} {Communication} {Theory}--{A} {Generalization} of {Information} {Theory}},
	volume = {21},
	issn = {0096-2244},
	url = {http://ieeexplore.ieee.org/document/1091610/},
	doi = {10.1109/TCOM.1973.1091610},
	abstract = {A generalization of informationtheory is presentedwith the aim of distinguishing the direction of information flow for mutually coupledstatistical systems. The bidirectionalcommunication theory refers totwo systems. Twodirectedtransinformationsaredefined which area measure of the statisticalcouplingbetween the systems. Their sum equals Shannon’s transinformation. An information flow diagram explains the relationbetween the directedtransinformations and theentropies of the sources. An extension to a group of such systems has also been proposed. The theory is able to describe the informational relationships between living beings and other multivariate complex systems as encountered in economy. An application example referring to groupbehavior with monkeys is given.},
	language = {en},
	number = {12},
	urldate = {2023-05-11},
	journal = {IEEE Transactions on Communications},
	author = {Marko, H.},
	month = dec,
	year = {1973},
	pages = {1345--1351},
	file = {Marko - 1973 - The Bidirectional Communication Theory--A Generali.pdf:/Users/jacquelinemaasch/Zotero/storage/LIIIS5N7/Marko - 1973 - The Bidirectional Communication Theory--A Generali.pdf:application/pdf},
}

@article{massey_causality_nodate,
	title = {{CAUSALITY}, {FEEDBACK} {AND} {DIRECTED} {INFORMATION}},
	abstract = {It is shown that the "usual definition" of a discrete memoryless channel (DMC) in fact prohibits the use of feedback. The difficulty stems from the confusion of causality and statistical dependence. An adequate definition of a DMC is given, as well as a definition of using a channel without feedback. A definition, closely based on an old idea of Marko, is given for the directed information flowing from one sequence to another. This directed information is used to give a simple proof of the well-known fact that the use of feedback cannot increase the capacity of a DMC. It is shown that, when feedback is present, directed information is a more useful quantity than the traditional mutual information.},
	language = {en},
	author = {Massey, James L},
	file = {Massey - CAUSALITY, FEEDBACK AND DIRECTED INFORMATION.pdf:/Users/jacquelinemaasch/Zotero/storage/VYCE2SVR/Massey - CAUSALITY, FEEDBACK AND DIRECTED INFORMATION.pdf:application/pdf},
}

@book{ay_information_2017,
	address = {Cham},
	series = {Ergebnisse der {Mathematik} und ihrer {Grenzgebiete} 34},
	title = {Information {Geometry}},
	volume = {64},
	isbn = {978-3-319-56477-7 978-3-319-56478-4},
	url = {http://link.springer.com/10.1007/978-3-319-56478-4},
	language = {en},
	urldate = {2023-05-11},
	publisher = {Springer International Publishing},
	author = {Ay, Nihat and Jost, Jürgen and Lê, Hông Vân and Schwachhöfer, Lorenz},
	year = {2017},
	doi = {10.1007/978-3-319-56478-4},
	file = {Ay et al. - 2017 - Information Geometry.pdf:/Users/jacquelinemaasch/Zotero/storage/DC3TSG8Y/Ay et al. - 2017 - Information Geometry.pdf:application/pdf},
}

@article{amblard_relation_2012,
	title = {The {Relation} between {Granger} {Causality} and {Directed} {Information} {Theory}: {A} {Review}},
	volume = {15},
	issn = {1099-4300},
	shorttitle = {The {Relation} between {Granger} {Causality} and {Directed} {Information} {Theory}},
	url = {http://www.mdpi.com/1099-4300/15/1/113},
	doi = {10.3390/e15010113},
	abstract = {This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The deﬁnitions of Granger causality based on prediction are recalled, and the importance of the observation set is discussed. We present the deﬁnitions based on conditional independence. The notion of instantaneous coupling is included in the deﬁnitions. The concept of Granger causality graphs is discussed. We present directed information theory from the perspective of studies of causal inﬂuences between stochastic processes. Causal conditioning appears to be the cornerstone for the relation between information theory and Granger causality. In the bivariate case, the fundamental measure is the directed information, which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling. We show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure, a relation known for the linear Gaussian case. We study the multivariate case, showing that the useful decomposition is blurred by instantaneous coupling. The links are further developed by studying how measures based on directed information theory naturally emerge from Granger causality inference frameworks as hypothesis testing.},
	language = {en},
	number = {1},
	urldate = {2023-05-11},
	journal = {Entropy},
	author = {Amblard, Pierre-Olivier and Michel, Olivier},
	month = dec,
	year = {2012},
	pages = {113--143},
	file = {Amblard and Michel - 2012 - The Relation between Granger Causality and Directe.pdf:/Users/jacquelinemaasch/Zotero/storage/39CH6MZW/Amblard and Michel - 2012 - The Relation between Granger Causality and Directe.pdf:application/pdf},
}

@article{ay_information_2008,
	title = {{INFORMATION} {FLOWS} {IN} {CAUSAL} {NETWORKS}},
	volume = {11},
	issn = {0219-5259, 1793-6802},
	url = {https://www.worldscientific.com/doi/abs/10.1142/S0219525908001465},
	doi = {10.1142/S0219525908001465},
	abstract = {We use a notion of causal independence based on intervention, which is a fundamental concept of the theory of causal networks, to define a measure for the strength of a causal effect. We call this measure "information flow" and compare it with known information flow measures such as transfer entropy.},
	language = {en},
	number = {01},
	urldate = {2023-05-11},
	journal = {Advances in Complex Systems},
	author = {Ay, Nihat and Polani, Daniel},
	month = feb,
	year = {2008},
	pages = {17--41},
	file = {Ay and Polani - 2008 - INFORMATION FLOWS IN CAUSAL NETWORKS.pdf:/Users/jacquelinemaasch/Zotero/storage/7GJM9VW7/Ay and Polani - 2008 - INFORMATION FLOWS IN CAUSAL NETWORKS.pdf:application/pdf},
}

@article{lizier_local_2008,
	title = {Local information transfer as a spatiotemporal filter for complex systems},
	volume = {77},
	issn = {1539-3755, 1550-2376},
	url = {https://link.aps.org/doi/10.1103/PhysRevE.77.026110},
	doi = {10.1103/PhysRevE.77.026110},
	language = {en},
	number = {2},
	urldate = {2023-05-11},
	journal = {Physical Review E},
	author = {Lizier, Joseph T. and Prokopenko, Mikhail and Zomaya, Albert Y.},
	month = feb,
	year = {2008},
	pages = {026110},
	file = {Lizier et al. - 2008 - Local information transfer as a spatiotemporal fil.pdf:/Users/jacquelinemaasch/Zotero/storage/CHDH95JS/Lizier et al. - 2008 - Local information transfer as a spatiotemporal fil.pdf:application/pdf},
}

@article{amblard_directed_2011,
	title = {On directed information theory and {Granger} causality graphs},
	volume = {30},
	issn = {0929-5313, 1573-6873},
	url = {http://link.springer.com/10.1007/s10827-010-0231-x},
	doi = {10.1007/s10827-010-0231-x},
	abstract = {Directed information theory deals with communication channels with feedback. When applied to networks, a natural extension based on causal conditioning is needed. We show here that measures built from directed information theory in networks can be used to assess Granger causality graphs of stochastic processes. We show that directed information theory includes measures such as the transfer entropy, and that it is the adequate information theoretic framework needed for neuroscience applications, such as connectivity inference problems.},
	language = {en},
	number = {1},
	urldate = {2023-05-12},
	journal = {Journal of Computational Neuroscience},
	author = {Amblard, Pierre-Olivier and Michel, Olivier J. J.},
	month = feb,
	year = {2011},
	pages = {7--16},
	file = {Amblard and Michel - 2011 - On directed information theory and Granger causali.pdf:/Users/jacquelinemaasch/Zotero/storage/44SSXJ6J/Amblard and Michel - 2011 - On directed information theory and Granger causali.pdf:application/pdf},
}

@article{surasinghe_geometry_2020,
	title = {On {Geometry} of {Information} {Flow} for {Causal} {Inference}},
	volume = {22},
	issn = {1099-4300},
	url = {https://www.mdpi.com/1099-4300/22/4/396},
	doi = {10.3390/e22040396},
	abstract = {Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other ﬁelds. This paper takes the perspective of information ﬂow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information ﬂow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information ﬂow. Therefore, contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write GeoCy→x. This avoids some of the boundedness issues that we show exist for the transfer entropy, Ty→x. We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the Hénon map example, and ﬁnally a real physiological example relating breathing and heart rate function.},
	language = {en},
	number = {4},
	urldate = {2023-05-16},
	journal = {Entropy},
	author = {Surasinghe, Sudam and Bollt, Erik M.},
	month = mar,
	year = {2020},
	pages = {396},
	file = {Surasinghe and Bollt - 2020 - On Geometry of Information Flow for Causal Inferen.pdf:/Users/jacquelinemaasch/Zotero/storage/IINKT5YX/Surasinghe and Bollt - 2020 - On Geometry of Information Flow for Causal Inferen.pdf:application/pdf},
}

@inproceedings{etesami_directed_2014,
	address = {Portland, OR, USA},
	title = {Directed {Information} {Graphs}: {A} generalization of {Linear} {Dynamical} {Graphs}},
	isbn = {978-1-4799-3274-0 978-1-4799-3272-6 978-1-4799-3271-9},
	shorttitle = {Directed {Information} {Graphs}},
	url = {http://ieeexplore.ieee.org/document/6859362/},
	doi = {10.1109/ACC.2014.6859362},
	abstract = {We study the relationship between Directed Information Graphs (DIG) and Linear Dynamical Graphs (LDG), both of which are graphical models where nodes represent scalar random processes. DIGs are based on directed information and represent the causal dynamics between processes in a stochastic system. LDGs capture causal dynamics but only in linear dynamical systems and there are Wiener ﬁltering to do so in a subset of LDGs. This study shows that the DIGs are generalized version of the LDGs and any strictly causal LDGs can be reconstructed through learning the corresponding DIGs.},
	language = {en},
	urldate = {2023-05-16},
	booktitle = {2014 {American} {Control} {Conference}},
	publisher = {IEEE},
	author = {Etesami, Jalal and Kiyavash, Negar},
	month = jun,
	year = {2014},
	pages = {2563--2568},
	file = {Etesami and Kiyavash - 2014 - Directed Information Graphs A generalization of L.pdf:/Users/jacquelinemaasch/Zotero/storage/CNP45RLF/Etesami and Kiyavash - 2014 - Directed Information Graphs A generalization of L.pdf:application/pdf},
}

@article{marx_formally_nodate,
	title = {Formally {Justifying} {MDL}-based {Inference} of {Cause} and {Effect}},
	abstract = {The algorithmic independence of conditionals, which postulates that the causal mechanism is algorithmically independent of the cause, has recently inspired many highly successful approaches to distinguish cause from effect given only observational data. Most popular among these is the idea to approximate algorithmic independence via two-part Minimum Description Length (MDL). Although intuitively sensible, the link between the original postulate and practical two-part MDL encodings has so far been left vague. In this work, we close this gap by deriving a two-part formulation of this postulate, in terms of Kolmogorov complexity, which directly links to practical MDL encodings. To close the cycle, we prove that this formulation leads on expectation to the same inference result as the original postulate.},
	language = {en},
	author = {Marx, Alexander and Vreeken, Jilles},
	file = {Marx and Vreeken - Formally Justifying MDL-based Inference of Cause a.pdf:/Users/jacquelinemaasch/Zotero/storage/IQD753LL/Marx and Vreeken - Formally Justifying MDL-based Inference of Cause a.pdf:application/pdf},
}

@incollection{bossert_identification_2018,
	address = {Cham},
	title = {Identification of {Causal} {Dependences} in {Gene} {Regulatory} {Networks} {Using} {Algorithmic} {Information} {Theory}},
	isbn = {978-3-319-54728-2 978-3-319-54729-9},
	url = {http://link.springer.com/10.1007/978-3-319-54729-9_14},
	abstract = {This project aims at analyzing the causal structure of genetic regulatory networks of stem cells of plants using novel causal inference techniques to be developed here. Known methods for causal inference from statistical data usually require a large number of samples. Our preliminary work shows that it is in principle possible to infer causal relations from sample size one if the variables are high-dimensional, since algorithmic information provides additional hints on causal directions. Recent advances in genomic methods have allowed the simultaneous quantiﬁcation of all genes in an organism. To identify the causal relation between individual transcripts, we will use inducible expression to analyze the effect of the homeodomain transcription factor WUSCHEL on the regulatory network of plant stem cell control. After appropriate clustering of the genes, we obtain a causal network between extremely high-dimensional variables, to which algorithmic information theory based methods can be applied. The inferred causal relation will then be tested by advanced experiments.},
	language = {en},
	urldate = {2023-05-16},
	booktitle = {Information- and {Communication} {Theory} in {Molecular} {Biology}},
	publisher = {Springer International Publishing},
	author = {Lohmann, Jan and Janzing, Dominik},
	editor = {Bossert, Martin},
	year = {2018},
	doi = {10.1007/978-3-319-54729-9_14},
	note = {Series Title: Lecture Notes in Bioengineering},
	pages = {315--330},
	file = {Lohmann and Janzing - 2018 - Identification of Causal Dependences in Gene Regul.pdf:/Users/jacquelinemaasch/Zotero/storage/FGXX2HFK/Lohmann and Janzing - 2018 - Identification of Causal Dependences in Gene Regul.pdf:application/pdf},
}

@incollection{guyon_cause-effect_2019,
	address = {Cham},
	title = {The {Cause}-{Effect} {Problem}: {Motivation}, {Ideas}, and {Popular} {Misconceptions}},
	isbn = {978-3-030-21809-6 978-3-030-21810-2},
	shorttitle = {The {Cause}-{Effect} {Problem}},
	url = {http://link.springer.com/10.1007/978-3-030-21810-2_1},
	abstract = {Telling cause from effect from observations of just two variables has attracted increasing interest since more than one decade. On the one hand, it deﬁnes a nice binary classiﬁcation problem for which it is easy to deﬁne a success rate, in contrast to more general causal inference tasks where no straightforward performance criteria exist. On the other hand, it fascinates researchers because solving this elementary task implies statistical asymmetries between cause and effect that were previously unknown. Discussing some real-world and toy examples, I argue that humans seem to have some intuition about these asymmetries, but I also argue that some straightforward ideas to distinguish between cause and effect are ﬂawed. The discussion on the origin of the true asymmetries relates machine learning, philosophy and physics. For instance, the postulate that Pcause and Peffect{\textbar}cause contain no information about each other (while Peffect and Pcause{\textbar}effect may satisfy some ‘nongeneric’ relations), is relevant for semi-supervised learning on the one hand, but is also related to the thermodynamic arrow of time on the other hand.},
	language = {en},
	urldate = {2023-05-16},
	booktitle = {Cause {Effect} {Pairs} in {Machine} {Learning}},
	publisher = {Springer International Publishing},
	author = {Janzing, Dominik},
	editor = {Guyon, Isabelle and Statnikov, Alexander and Batu, Berna Bakir},
	year = {2019},
	doi = {10.1007/978-3-030-21810-2_1},
	note = {Series Title: The Springer Series on Challenges in Machine Learning},
	pages = {3--26},
	file = {Janzing - 2019 - The Cause-Effect Problem Motivation, Ideas, and P.pdf:/Users/jacquelinemaasch/Zotero/storage/IIZQXXXU/Janzing - 2019 - The Cause-Effect Problem Motivation, Ideas, and P.pdf:application/pdf},
}

@article{zhang_adjusting_2022,
	title = {Adjusting for indirectly measured confounding using large-scale propensity score},
	volume = {134},
	issn = {15320464},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S153204642200209X},
	doi = {10.1016/j.jbi.2022.104204},
	abstract = {Confounding remains one of the major challenges to causal inference with observational data. This problem is paramount in medicine, where we would like to answer causal questions from large observational datasets like electronic health records (EHRs) and administrative claims. Modern medical data typically contain tens of thousands of covariates. Such a large set carries hope that many of the confounders are directly measured, and further hope that others are indirectly measured through their correlation with measured covariates. How can we exploit these large sets of covariates for causal inference? To help answer this question, this paper examines the performance of the large-scale propensity score (LSPS) approach on causal analysis of medical data. We demonstrate that LSPS may adjust for indirectly measured confounders by including tens of thousands of covariates that may be correlated with them. We present conditions under which LSPS removes bias due to indirectly measured confounders, and we show that LSPS may avoid bias when inadvertently adjusting for variables (like colliders) that otherwise can induce bias. We demonstrate the performance of LSPS with both simulated medical data and real medical data.},
	language = {en},
	urldate = {2023-05-22},
	journal = {Journal of Biomedical Informatics},
	author = {Zhang, Linying and Wang, Yixin and Schuemie, Martijn J. and Blei, David M. and Hripcsak, George},
	month = oct,
	year = {2022},
	pages = {104204},
	file = {Zhang et al. - 2022 - Adjusting for indirectly measured confounding usin.pdf:/Users/jacquelinemaasch/Zotero/storage/ZRQV3BRR/Zhang et al. - 2022 - Adjusting for indirectly measured confounding usin.pdf:application/pdf},
}

@article{thoemmes_m-bias_2015,
	title = {M-bias, {Butterfly} {Bias}, and {Butterfly} {Bias} with {Correlated} {Causes} – {A} {Comment} on {Ding} and {Miratrix} (2015)},
	volume = {3},
	issn = {2193-3677, 2193-3685},
	url = {https://www.degruyter.com/document/doi/10.1515/jci-2015-0012/html},
	doi = {10.1515/jci-2015-0012},
	abstract = {Ding and Miratrix [1] recently concluded that adjustment on a pre-treatment covariate is almost always preferable to reduce bias. I extend the examined parameter space of the models considered by Ding and Miratrix, and consider slight extensions of their models as well. Similar to the conclusion by Pearl [7], I identify constellations in which bias due to adjustment, or failing to adjust is symmetrical, but also confirm some findings of Ding and Miratrix.},
	language = {en},
	number = {2},
	urldate = {2023-06-06},
	journal = {Journal of Causal Inference},
	author = {Thoemmes, Felix},
	month = sep,
	year = {2015},
	pages = {253--258},
	file = {Thoemmes - 2015 - M-bias, Butterfly Bias, and Butterfly Bias with Co.pdf:/Users/jacquelinemaasch/Zotero/storage/TD4L4GDI/Thoemmes - 2015 - M-bias, Butterfly Bias, and Butterfly Bias with Co.pdf:application/pdf},
}


@article{pearl_linear_2013,
	title = {Linear {Models}: {A} {Useful} “{Microscope}” for {Causal} {Analysis}},
	volume = {1},
	abstract = {This note reviews basic techniques of linear path analysis and demonstrates, using simple examples, how causal phenomena of non-trivial character can be understood, exempliﬁed and analyzed using diagrams and a few algebraic steps. The techniques allow for swift assessment of how various features of the model impact the phenomenon under investigation. This includes: Simpson’s paradox, case-control bias, selection bias, collider bias, reverse regression, bias ampliﬁcation, near instruments, and measurement errors.},
	language = {en},
	number = {1},
	journal = {Journal of Causal Inference},
	author = {Pearl, Judea},
	year = {2013},
	pages = {155--170},
	file = {Pearl - 2013 - Linear Models A Useful “Microscope” for Causal An.pdf:/Users/jacquelinemaasch/Zotero/storage/TBPSJTTW/Pearl - 2013 - Linear Models A Useful “Microscope” for Causal An.pdf:application/pdf},
}


@article{pearl_comment_2015,
	title = {Comment on {Ding} and {Miratrix}: “{To} {Adjust} or {Not} to {Adjust}?”},
	volume = {3},
	issn = {2193-3677, 2193-3685},
	shorttitle = {Comment on {Ding} and {Miratrix}},
	url = {https://www.degruyter.com/document/doi/10.1515/jci-2015-0004/html},
	doi = {10.1515/jci-2015-0004},
	language = {en},
	number = {1},
	urldate = {2023-06-06},
	journal = {Journal of Causal Inference},
	author = {Pearl, Judea},
	month = mar,
	year = {2015},
	pages = {59--60},
	file = {Pearl - 2015 - Comment on Ding and Miratrix “To Adjust or Not to.pdf:/Users/jacquelinemaasch/Zotero/storage/222QS6FK/Pearl - 2015 - Comment on Ding and Miratrix “To Adjust or Not to.pdf:application/pdf},
}

@misc{ding_adjust_2014,
	title = {To {Adjust} or {Not} to {Adjust}? {Sensitivity} {Analysis} of {M}-{Bias} and {Butterfly}-{Bias}},
	shorttitle = {To {Adjust} or {Not} to {Adjust}?},
	url = {http://arxiv.org/abs/1408.0324},
	abstract = {M-Bias,” as it is called in the epidemiologic literature, is the bias introduced by conditioning on a pretreatment covariate due to a particular “M-Structure” between two latent factors, an observed treatment, an outcome, and a “collider.” This potential source of bias, which can occur even when the treatment and the outcome are not confounded, has been a source of considerable controversy. We here present formulae for identifying under which circumstances biases are inﬂated or reduced. In particular, we show that the magnitude of M-Bias in linear structural equation models tends to be relatively small compared to confounding bias, suggesting that it is generally not a serious concern in many applied settings. These theoretical results are consistent with recent empirical ﬁndings from simulation studies. We also generalize the M-Bias setting (1) to allow for the correlation between the latent factors to be nonzero, and (2) to allow for the collider to be a confounder between the treatment and the outcome. These results demonstrate that mild deviations from the M-Structure tend to increase confounding bias more rapidly than M-Bias, suggesting that choosing to condition on any given covariate is generally the superior choice. As an application, we re-examine a controversial example between Professors Donald Rubin and Judea Pearl.},
	language = {en},
	urldate = {2023-06-06},
	publisher = {arXiv},
	author = {Ding, Peng and Miratrix, Luke},
	month = aug,
	year = {2014},
	note = {arXiv:1408.0324 [math, stat]},
	keywords = {Mathematics - Statistics Theory},
	file = {Ding and Miratrix - 2014 - To Adjust or Not to Adjust Sensitivity Analysis o.pdf:/Users/jacquelinemaasch/Zotero/storage/FTPJSLYX/Ding and Miratrix - 2014 - To Adjust or Not to Adjust Sensitivity Analysis o.pdf:application/pdf},
}

@misc{chen_correcting_2022,
	title = {Correcting {Confounding} via {Random} {Selection} of {Background} {Variables}},
	url = {http://arxiv.org/abs/2202.02150},
	abstract = {We propose a method to distinguish causal influence from hidden confounding in the following scenario: given a target variable Y, potential causal drivers X, and a large number of background features, we propose a novel criterion for identifying causal relationship based on the stability of regression coefficients of X on Y with respect to selecting different background features. To this end, we propose a statistic V measuring the coefficient's variability. We prove, subject to a symmetry assumption for the background influence, that V converges to zero if and only if X contains no causal drivers. In experiments with simulated data, the method outperforms state of the art algorithms. Further, we report encouraging results for real-world data. Our approach aligns with the general belief that causal insights admit better generalization of statistical associations across environments, and justifies similar existing heuristic approaches from the literature.},
	language = {en},
	urldate = {2023-06-06},
	publisher = {arXiv},
	author = {Chen, You-Lin and Minorics, Lenon and Janzing, Dominik},
	month = feb,
	year = {2022},
	note = {arXiv:2202.02150 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Chen et al. - 2022 - Correcting Confounding via Random Selection of Bac.pdf:/Users/jacquelinemaasch/Zotero/storage/LXNUP78H/Chen et al. - 2022 - Correcting Confounding via Random Selection of Bac.pdf:application/pdf},
}

@article{zhang_computational_2022,
	title = {Computational causal discovery: {Advantages} and assumptions},
	volume = {37},
	issn = {2171-679X, 0495-4548},
	shorttitle = {Computational causal discovery},
	url = {https://ojs.ehu.eus/index.php/THEORIA/article/view/22904},
	doi = {10.1387/theoria.22904},
	abstract = {I would like to congratulate James Woodward for another landmark accomplishment, after publishing his Making Things Happen: A Theory of Causal Explanation (Woodward, 2003). Makes Things Happens gives an elegant interventionist theory for understanding explanation and causation. The new contribution (W­ oodward, 2022) relies on that theory and further makes a big step towards empirical inference of causal relations from nonexperimental data. In this paper, I will focus on some of the emerging computational methods for finding causal relations from non-experimental data and attempt to complement Woodward’s contribution with discussions on 1) how these methods are connected to the interventionist theory of causality, 2) how informative the output of the methods is, including whether they output directed causal graphs and how they deal with confounders (unmeasured common causes of two measured variables), and 3) the assumptions underlying the asymptotic correctness of the output of the methods about causal relations. Different causal discovery methods may rely on different aspects of the joint distribution of the data, and this discussion aims to provide a technical account of the assumptions.},
	language = {en},
	number = {1},
	urldate = {2023-06-06},
	journal = {THEORIA. An International Journal for Theory, History and Foundations of Science},
	author = {Zhang, Kun},
	month = apr,
	year = {2022},
	file = {Zhang - 2022 - Computational causal discovery Advantages and ass.pdf:/Users/jacquelinemaasch/Zotero/storage/R5SH9M4S/Zhang - 2022 - Computational causal discovery Advantages and ass.pdf:application/pdf},
}

@misc{scholkopf_statistical_2022,
	title = {From {Statistical} to {Causal} {Learning}},
	url = {http://arxiv.org/abs/2204.00607},
	abstract = {We describe basic ideas underlying research to build and understand artificially intelligent systems: from symbolic approaches via statistical learning to interventional models relying on concepts of causality. Some of the hard open problems of machine learning and AI are intrinsically related to causality, and progress may require advances in our understanding of how to model and infer causality from data.},
	language = {en},
	urldate = {2023-06-06},
	publisher = {arXiv},
	author = {Schölkopf, Bernhard and von Kügelgen, Julius},
	month = apr,
	year = {2022},
	note = {arXiv:2204.00607 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Schölkopf and von Kügelgen - 2022 - From Statistical to Causal Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/R7TUBQAG/Schölkopf and von Kügelgen - 2022 - From Statistical to Causal Learning.pdf:application/pdf},
}

@article{rotnitzky_ecient_nodate,
	title = {Eﬃcient {Adjustment} {Sets} for {Population} {Average} {Causal} {Treatment} {Eﬀect} {Estimation} in {Graphical} {Models}},
	abstract = {The method of covariate adjustment is often used for estimation of total treatment effects from observational studies. Restricting attention to causal linear models, a recent article (Henckel et al., 2019) derived two novel graphical criteria: one to compare the asymptotic variance of linear regression treatment eﬀect estimators that control for certain distinct adjustment sets and another to identify the optimal adjustment set that yields the least squares estimator with the smallest asymptotic variance. In this paper we show that the same graphical criteria can be used in non-parametric causal graphical models when treatment eﬀects are estimated using non-parametrically adjusted estimators of the interventional means. We also provide a new graphical criterion for determining the optimal adjustment set among the minimal adjustment sets and another novel graphical criterion for comparing time dependent adjustment sets. We show that uniformly optimal time dependent adjustment sets do not always exist. For point interventions, we provide a sound and complete graphical criterion for determining when a non-parametric optimally adjusted estimator of an interventional mean, or of a contrast of interventional means, is semiparametric eﬃcient under the non-parametric causal graphical model. In addition, when the criterion is not met, we provide a sound algorithm that checks for possible simpliﬁcations of the eﬃcient inﬂuence function of the parameter. Finally, we ﬁnd an interesting connection between identiﬁcation and eﬃcient covariate adjustment estimation. Speciﬁcally, we show that if there exists an identifying formula for an interventional mean that depends only on treatment, outcome and mediators, then the non-parametric optimally adjusted estimator can never be globally eﬃcient under the causal graphical model.},
	language = {en},
	author = {Rotnitzky, Andrea and Smucler, Ezequiel},
	file = {Rotnitzky and Smucler - Eﬃcient Adjustment Sets for Population Average Cau.pdf:/Users/jacquelinemaasch/Zotero/storage/33EUAPZR/Rotnitzky and Smucler - Eﬃcient Adjustment Sets for Population Average Cau.pdf:application/pdf},
}

@article{perkovic_complete_2018,
	title = {Complete {Graphical} {Characterization} and {Construction} of {Adjustment} {Sets} in {Markov} {Equivalence} {Classes} of {Ancestral} {Graphs}},
	volume = {18},
	abstract = {We present a graphical criterion for covariate adjustment that is sound and complete for four diﬀerent classes of causal graphical models: directed acyclic graphs (DAGs), maximal ancestral graphs (MAGs), completed partially directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our criterion uniﬁes covariate adjustment for a large set of graph classes. Moreover, we deﬁne an explicit set that satisﬁes our criterion, if there is any set that satisﬁes our criterion. We also give eﬃcient algorithms for constructing all sets that fulﬁll our criterion, implemented in the R package dagitty. Finally, we discuss the relationship between our criterion and other criteria for adjustment, and we provide new soundness and completeness proofs for the adjustment criterion for DAGs.},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Perković, Emilija and Textor, Johannes and Kalisch, Markus},
	year = {2018},
	pages = {1--62},
	file = {Perković et al. - Complete Graphical Characterization and Constructi.pdf:/Users/jacquelinemaasch/Zotero/storage/HDSF33GT/Perković et al. - Complete Graphical Characterization and Constructi.pdf:application/pdf},
}

@article{van_der_zander_separators_2019,
	title = {Separators and adjustment sets in causal graphs: {Complete} criteria and an algorithmic framework},
	volume = {270},
	issn = {00043702},
	shorttitle = {Separators and adjustment sets in causal graphs},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0004370219300025},
	doi = {10.1016/j.artint.2018.12.006},
	language = {en},
	urldate = {2023-06-06},
	journal = {Artificial Intelligence},
	author = {Van Der Zander, Benito and Liśkiewicz, Maciej and Textor, Johannes},
	month = may,
	year = {2019},
	pages = {1--40},
	file = {Van Der Zander et al. - 2019 - Separators and adjustment sets in causal graphs C.pdf:/Users/jacquelinemaasch/Zotero/storage/E2EF6FZR/Van Der Zander et al. - 2019 - Separators and adjustment sets in causal graphs C.pdf:application/pdf},
}

@article{gan_causal_nodate,
	title = {Causal {Inference} with {Selectively} {Deconfounded} {Data}},
	abstract = {In general, when treatments and eﬀects are observed, but confounders are not, the average treatment eﬀect (ATE) is not identiﬁable. To estimate the ATE, a practitioner must then either (a) collect deconfounded data; (b) run a clinical trial; or (c) elucidate further properties of the causal graph that might render the ATE identiﬁable. In this paper, we consider the beneﬁt of incorporating a large confounded observational dataset (confounder unobserved ) alongside a small deconfounded observational dataset (confounder revealed ) when estimating the ATE. Our theoretical results show that the inclusion of confounded data can signiﬁcantly reduce the quantity of deconfounded data required to estimate the ATE to within a desired accuracy level. Moreover, in some cases—say, genetics—we could imagine retrospectively selecting samples to deconfound. We demonstrate that by actively selecting these samples based upon the (already observed) treatment and outcome, we can reduce our data dependence further. Our theoretical results establish that the worstcase relative performance of our approach (vs. random selection) is bounded while our best-case gains are unbounded. We perform extensive synthetic experiments to validate our theoretical results. Finally, we demonstrate the practical beneﬁts of selective deconfounding using a large real-world dataset related to genetic mutation in cancer.},
	language = {en},
	author = {Gan, Kyra and Li, Andrew A and Lipton, Zachary C and Tayur, Sridhar},
	file = {Gan et al. - Causal Inference with Selectively Deconfounded Dat.pdf:/Users/jacquelinemaasch/Zotero/storage/C62S998N/Gan et al. - Causal Inference with Selectively Deconfounded Dat.pdf:application/pdf},
}

@misc{immer_identifiability_2023,
	title = {On the {Identifiability} and {Estimation} of {Causal} {Location}-{Scale} {Noise} {Models}},
	url = {http://arxiv.org/abs/2210.09054},
	abstract = {We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f (X) + g(X)N . Despite the generality of the model class, we show the causal direction is identiﬁable up to some pathological cases. To empirically validate these theoretical ﬁndings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on probabilistic neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. Since the neural network approach can ﬁt functions of arbitrary complexity, it has an edge over the feature map-based approach in terms of empirical performance. When the feature maps are correctly speciﬁed, however, we can prove that our estimator is jointly concave, which allows us to derive stronger guarantees for the cause-effect identiﬁcation task.},
	language = {en},
	urldate = {2023-06-06},
	publisher = {arXiv},
	author = {Immer, Alexander and Schultheiss, Christoph and Vogt, Julia E. and Schölkopf, Bernhard and Bühlmann, Peter and Marx, Alexander},
	month = jun,
	year = {2023},
	note = {arXiv:2210.09054 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Immer et al. - 2023 - On the Identifiability and Estimation of Causal Lo.pdf:/Users/jacquelinemaasch/Zotero/storage/SRJ4B7PB/Immer et al. - 2023 - On the Identifiability and Estimation of Causal Lo.pdf:application/pdf},
}

@article{kuang_ivy_nodate,
	title = {Ivy: {Instrumental} {Variable} {Synthesis} for {Causal} {Inference}},
	abstract = {A popular way to estimate the causal eﬀect of a variable x on y from observational data is to use an instrumental variable (IV): a third variable z that aﬀects y only through x. The more strongly z is associated with x, the more reliable the estimate is, but such strong IVs are diﬃcult to ﬁnd. Instead, practitioners combine more commonly available IV candidates—which are not necessarily strong, or even valid, IVs—into a single “summary” that is plugged into causal effect estimators in place of an IV. In genetic epidemiology, such approaches are known as allele scores. Allele scores require strong assumptions—independence and validity of all IV candidates—for the resulting estimate to be reliable. To relax these assumptions, we propose Ivy, a new method to combine IV candidates that can handle correlated and invalid IV candidates in a robust manner. Theoretically, we characterize this robustness, its limits, and its impact on the resulting causal estimates. Empirically, we show that Ivy can correctly identify the directionality of known relationships and is robust against false discovery (median eﬀect size ≤ 0.025) on three real-world datasets with no causal eﬀects, while allele scores return more biased estimates (median eﬀect size ≥ 0.118).},
	language = {en},
	author = {Kuang, Zhaobin and Sala, Frederic and Sohoni, Nimit and Wu, Sen and Cordova-Palomera, Aldo and Dunnmon, Jared and Priest, James and Re, Christopher},
	file = {Kuang et al. - Ivy Instrumental Variable Synthesis for Causal In.pdf:/Users/jacquelinemaasch/Zotero/storage/3T5JFUVS/Kuang et al. - Ivy Instrumental Variable Synthesis for Causal In.pdf:application/pdf},
}

@article{cole_illustrating_2010,
	title = {Illustrating bias due to conditioning on a collider},
	volume = {39},
	issn = {1464-3685, 0300-5771},
	url = {https://academic.oup.com/ije/article-lookup/doi/10.1093/ije/dyp334},
	doi = {10.1093/ije/dyp334},
	language = {en},
	number = {2},
	urldate = {2023-06-06},
	journal = {International Journal of Epidemiology},
	author = {Cole, Stephen R and Platt, Robert W and Schisterman, Enrique F and Chu, Haitao and Westreich, Daniel and Richardson, David and Poole, Charles},
	month = apr,
	year = {2010},
	pages = {417--420},
	file = {Cole et al. - 2010 - Illustrating bias due to conditioning on a collide.pdf:/Users/jacquelinemaasch/Zotero/storage/M98U3HYM/Cole et al. - 2010 - Illustrating bias due to conditioning on a collide.pdf:application/pdf},
}

@article{scholkopf_causal_nodate,
	title = {On {Causal} and {Anticausal} {Learning}},
	abstract = {We consider the problem of function estimation in the case where an underlying causal model can be inferred. This has implications for popular scenarios such as covariate shift, concept drift, transfer learning and semi-supervised learning. We argue that causal knowledge may facilitate some approaches for a given problem, and rule out others. In particular, we formulate a hypothesis for when semi-supervised learning can help, and corroborate it with empirical results.},
	language = {en},
	author = {Schölkopf, Bernhard and Janzing, Dominik and Peters, Jonas and Sgouritsa, Eleni and Zhang, Kun and Mooij, Joris},
	file = {Schölkopf et al. - On Causal and Anticausal Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/SBS5HNLK/Schölkopf et al. - On Causal and Anticausal Learning.pdf:application/pdf},
}

@article{janzing_semi-supervised_nodate,
	title = {Semi-{Supervised} {Interpolation} in an {Anticausal} {Learning} {Scenario}},
	abstract = {According to a recently stated ‘independence postulate’, the distribution Pcause contains no information about the conditional Peﬀect{\textbar}cause while Peﬀect may contain information about Pcause{\textbar}eﬀect. Since semi-supervised learning (SSL) attempts to exploit information from PX to assist in predicting Y from X, it should only work in anticausal direction, i.e., when Y is the cause and X is the eﬀect. In causal direction, when X is the cause and Y the eﬀect, unlabelled x-values should be useless. To shed light on this asymmetry, we study a deterministic causal relation Y = f (X) as recently assayed in InformationGeometric Causal Inference (IGCI). Within this model, we discuss two options to formalize the independence of PX and f as an orthogonality of vectors in appropriate inner product spaces. We prove that unlabelled data help for the problem of interpolating a monotonically increasing function if and only if the orthogonality conditions are violated – which we only expect for the anticausal direction. Here, performance of SSL and its supervised baseline analogue is measured in terms of two diﬀerent loss functions: ﬁrst, the mean squared error and second the surprise in a Bayesian prediction scenario.},
	language = {en},
	author = {Janzing, Dominik and Scholkopf, Bernhard},
	file = {Janzing and Scholkopf - Semi-Supervised Interpolation in an Anticausal Lea.pdf:/Users/jacquelinemaasch/Zotero/storage/YCIVFFH4/Janzing and Scholkopf - Semi-Supervised Interpolation in an Anticausal Lea.pdf:application/pdf},
}

@article{cai_triad_nodate,
	title = {Triad {Constraints} for {Learning} {Causal} {Structure} of {Latent} {Variables}},
	abstract = {Learning causal structure from observational data has attracted much attention, and it is notoriously challenging to ﬁnd the underlying structure in the presence of confounders (hidden direct common causes of two variables). In this paper, by properly leveraging the non-Gaussianity of the data, we propose to estimate the structure over latent variables with the so-called Triad constraints: we design a form of "pseudo-residual" from three variables, and show that when causal relations are linear and noise terms are non-Gaussian, the causal direction between the latent variables for the three observed variables is identiﬁable by checking a certain kind of independence relationship. In other words, the Triad constraints help us to locate latent confounders and determine the causal direction between them. This goes far beyond the Tetrad constraints and reveals more information about the underlying structure from non-Gaussian data. Finally, based on the Triad constraints, we develop a two-step algorithm to learn the causal structure corresponding to measurement models. Experimental results on both synthetic and real data demonstrate the effectiveness and reliability of our method.},
	language = {en},
	author = {Cai, Ruichu and Xie, Feng and Glymour, Clark and Hao, Zhifeng and Zhang, Kun},
	file = {Cai et al. - Triad Constraints for Learning Causal Structure of.pdf:/Users/jacquelinemaasch/Zotero/storage/T46LK9R5/Cai et al. - Triad Constraints for Learning Causal Structure of.pdf:application/pdf},
}

@article{flanders_limits_2019,
	title = {Limits for the {Magnitude} of {M}-bias and {Certain} {Other} {Types} of {Structural} {Selection} {Bias}},
	volume = {30},
	issn = {1044-3983},
	url = {https://journals.lww.com/00001648-201907000-00006},
	doi = {10.1097/EDE.0000000000001031},
	abstract = {Background: Structural selection bias and confounding are key threats to validity of causal effect estimation. Here, we consider M-bias, a type of selection bias, described by Hernán et al as a situation wherein bias is caused by selecting on a variable that is caused by two other variables, one a cause of the exposure, the other a cause of the outcome. Our goals are to derive a bound for (the maximum) M-bias, explore through examples the magnitude of M-bias, illustrate how to apply the bound for other types of selection bias, and provide a program for directly calculating M-bias and the bound.
Methods: We derive a bound for selection bias assuming specific, causal relationships that characterize M-bias and further evaluate it using simulations.
Results: Through examples, we show that, in many plausible situations, M-bias will tend to be small. In some examples, the bias is not small–but plausibility of the examples, ultimately to be judged by the researcher, may be low. The examples also show how the M-bias bound yields bounds for other types of selection bias and also for confounding. The latter illustrates how Lee’s bound for confounding can arise as a limiting case of ours.
Conclusions: We have derived a new bound for M-bias. Examples illustrate how to apply it with other types of selection bias. They also show that it can yield tighter bounds in certain situations than a previously published bound for M-bias. Our examples suggest that M-bias may often, but not uniformly, be small.},
	language = {en},
	number = {4},
	urldate = {2023-06-06},
	journal = {Epidemiology},
	author = {Flanders, W. Dana and Ye, Dongni},
	month = jul,
	year = {2019},
	pages = {501--508},
	file = {Flanders and Ye - 2019 - Limits for the Magnitude of M-bias and Certain Oth.pdf:/Users/jacquelinemaasch/Zotero/storage/V8RT4RMH/Flanders and Ye - 2019 - Limits for the Magnitude of M-bias and Certain Oth.pdf:application/pdf},
}

@article{greenland_quantifying_2003,
	title = {Quantifying {Biases} in {Causal} {Models}: {Classical} {Confounding} vs {Collider}-{Stratification} {Bias}},
	volume = {14},
	issn = {1044-3983},
	shorttitle = {Quantifying {Biases} in {Causal} {Models}},
	url = {http://journals.lww.com/00001648-200305000-00009},
	doi = {10.1097/01.EDE.0000042804.12056.6C},
	language = {en},
	number = {3},
	urldate = {2023-06-06},
	journal = {Epidemiology},
	author = {Greenland, Sander},
	month = may,
	year = {2003},
	pages = {300--306},
	file = {Greenland - 2003 - Quantifying Biases in Causal Models Classical Con.pdf:/Users/jacquelinemaasch/Zotero/storage/C25W3YJV/Greenland - 2003 - Quantifying Biases in Causal Models Classical Con.pdf:application/pdf},
}

@article{ding_sensitivity_2016,
	title = {Sensitivity {Analysis} {Without} {Assumptions}:},
	volume = {27},
	issn = {1044-3983},
	shorttitle = {Sensitivity {Analysis} {Without} {Assumptions}},
	url = {http://journals.lww.com/00001648-201605000-00011},
	doi = {10.1097/EDE.0000000000000457},
	abstract = {Unmeasured confounding may undermine the validity of causal inference with observational studies. Sensitivity analysis provides an attractive way to partially circumvent this issue by assessing the potential influence of unmeasured confounding on causal conclusions. However, previous sensitivity analysis approaches often make strong and untestable assumptions such as having an unmeasured confounder that is binary, or having no interaction between the effects of the exposure and the confounder on the outcome, or having only one unmeasured confounder. Without imposing any assumptions on the unmeasured confounder or confounders, we derive a bounding factor and a sharp inequality such that the sensitivity analysis parameters must satisfy the inequality if an unmeasured confounder is to explain away the observed effect estimate or reduce it to a particular level. Our approach is easy to implement and involves only two sensitivity parameters. Surprisingly, our bounding factor, which makes no simplifying assumptions, is no more conservative than a number of previous sensitivity analysis techniques that do make assumptions. Our new bounding factor implies not only the traditional Cornfield conditions that both the relative risk of the exposure on the confounder and that of the confounder on the outcome must satisfy but also a high threshold that the maximum of these relative risks must satisfy. Furthermore, this new bounding factor can be viewed as a measure of the strength of confounding between the exposure and the outcome induced by a confounder.},
	language = {en},
	number = {3},
	urldate = {2023-06-06},
	journal = {Epidemiology},
	author = {Ding, Peng and VanderWeele, Tyler J.},
	month = may,
	year = {2016},
	pages = {368--377},
	file = {Ding and VanderWeele - 2016 - Sensitivity Analysis Without Assumptions.pdf:/Users/jacquelinemaasch/Zotero/storage/KQ5UKNT5/Ding and VanderWeele - 2016 - Sensitivity Analysis Without Assumptions.pdf:application/pdf},
}

@article{shpitser_validity_2010,
	title = {On the {Validity} of {Covariate} {Adjustment} for {Estimating} {Causal} {Eﬀects}},
	abstract = {Identifying eﬀects of actions (treatments) on outcome variables from observational data and causal assumptions is a fundamental problem in causal inference. This identiﬁcation is made diﬃcult by the presence of confounders which can be related to both treatment and outcome variables. Confounders are often handled, both in theory and in practice, by adjusting for covariates, in other words considering outcomes conditioned on treatment and covariate values, weighed by probability of observing those covariate values. In this paper, we give a complete graphical criterion for covariate adjustment, which we term the adjustment criterion, and derive some interesting corollaries of the completeness of this criterion.},
	language = {en},
	journal = {Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence},
	author = {Shpitser, Ilya and VanderWeele, Tyler and Robins, James M},
	year = {2010},
	pages = {527--536},
	file = {Shpitser et al. - On the Validity of Covariate Adjustment for Estima.pdf:/Users/jacquelinemaasch/Zotero/storage/WX8NE3XC/Shpitser et al. - On the Validity of Covariate Adjustment for Estima.pdf:application/pdf},
}

@article{perkovic_complete_2015,
	title = {A {Complete} {Generalized} {Adjustment} {Criterion}},
	abstract = {Covariate adjustment is a widely used approach to estimate total causal effects from observational data. Several graphical criteria have been developed in recent years to identify valid covariates for adjustment from graphical causal models. These criteria can handle multiple causes, latent confounding, or partial knowledge of the causal structure; however, their diversity is confusing and some of them are only sufﬁcient, but not necessary. In this paper, we present a criterion that is necessary and sufﬁcient for four different classes of graphical causal models: directed acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our criterion subsumes the existing ones and in this way uniﬁes adjustment set construction for a large set of graph classes.},
	language = {en},
	journal = {Proceedings of the Thirty-First Conference of Uncertainty in Artificial Intelligence (UAI)},
	author = {Perkovic, Emilija and Textor, Johannes and Kalisch, Markus and Maathuis, Marloes H},
	year = {2015},
	pages = {682--691},
	file = {Perkovic et al. - A Complete Generalized Adjustment Criterion.pdf:/Users/jacquelinemaasch/Zotero/storage/IPHI6ZAG/Perkovic et al. - A Complete Generalized Adjustment Criterion.pdf:application/pdf},
}

@article{loh_data-driven_2023,
	title = {Data-driven covariate selection for confounding adjustment by focusing on the stability of the effect estimator.},
	issn = {1939-1463, 1082-989X},
	url = {http://doi.apa.org/getdoi.cfm?doi=10.1037/met0000564},
	doi = {10.1037/met0000564},
	abstract = {Valid inference of cause-and-effect relations in observational studies necessitates adjusting for common causes of the focal predictor (i.e., treatment) and the outcome. When such common causes, henceforth termed confounders, remain unadjusted for, they generate spurious correlations that lead to biased causal effect estimates. But routine adjustment for all available covariates, when only a subset are truly confounders, is known to yield potentially inefﬁcient and unstable estimators. In this article, we introduce a datadriven confounder selection strategy that focuses on stable estimation of the treatment effect. The approach exploits the causal knowledge that after adjusting for confounders to eliminate all confounding biases, adding any remaining non-confounding covariates associated with only treatment or outcome, but not both, should not systematically change the effect estimator. The strategy proceeds in two steps. First, we prioritize covariates for adjustment by probing how strongly each covariate is associated with treatment and outcome. Next, we gauge the stability of the effect estimator by evaluating its trajectory adjusting for different covariate subsets. The smallest subset that yields a stable effect estimate is then selected. Thus, the strategy offers direct insight into the (in)sensitivity of the effect estimator to the chosen covariates for adjustment. The ability to correctly select confounders and yield valid causal inferences following data-driven covariate selection is evaluated empirically using extensive simulation studies. Furthermore, we compare the introduced method empirically with routine variable selection methods. Finally, we demonstrate the procedure using two publicly available real-world datasets. A step-by-step practical guide with user-friendly R functions is included.},
	language = {en},
	urldate = {2023-06-09},
	journal = {Psychological Methods},
	author = {Loh, Wen Wei and Ren, Dongning},
	month = apr,
	year = {2023},
	file = {Loh and Ren - 2023 - Data-driven covariate selection for confounding ad.pdf:/Users/jacquelinemaasch/Zotero/storage/IW7IHN2F/Loh and Ren - 2023 - Data-driven covariate selection for confounding ad.pdf:application/pdf},
}

@article{haggstrom_rejoinder_2018,
	title = {Rejoinder to {Discussions} on: {Data}-driven confounder selection via {Markov} and {Bayesian} networks: {Rejoinder} to {Discussions} on: {Data}-{Driven} {Confounder} {Selection} via {Markov} and {Bayesian} {Networks}},
	volume = {74},
	issn = {0006341X},
	shorttitle = {Rejoinder to {Discussions} on},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12783},
	doi = {10.1111/biom.12783},
	language = {en},
	number = {2},
	urldate = {2023-06-09},
	journal = {Biometrics},
	author = {Häggström, Jenny},
	month = jun,
	year = {2018},
	pages = {407--410},
	file = {Häggström - 2018 - Rejoinder to Discussions on Data-driven confounde.pdf:/Users/jacquelinemaasch/Zotero/storage/XHL9V2BN/Häggström - 2018 - Rejoinder to Discussions on Data-driven confounde.pdf:application/pdf},
}

@article{dorie_automated_2019,
	title = {Automated versus {Do}-{It}-{Yourself} {Methods} for {Causal} {Inference}: {Lessons} {Learned} from a {Data} {Analysis} {Competition}},
	volume = {34},
	issn = {0883-4237},
	shorttitle = {Automated versus {Do}-{It}-{Yourself} {Methods} for {Causal} {Inference}},
	url = {https://projecteuclid.org/journals/statistical-science/volume-34/issue-1/Automated-versus-Do-It-Yourself-Methods-for-Causal-Inference/10.1214/18-STS667.full},
	doi = {10.1214/18-STS667},
	abstract = {Statisticians have made great progress in creating methods that reduce our reliance on parametric assumptions. However, this explosion in research has resulted in a breadth of inferential strategies that both create opportunities for more reliable inference as well as complicate the choices that an applied researcher has to make and defend. Relatedly, researchers advocating for new methods typically compare their method to at best 2 or 3 other causal inference strategies and test using simulations that may or may not be designed to equally tease out ﬂaws in all the competing methods. The causal inference data analysis challenge, “Is Your SATT Where It’s At?”, launched as part of the 2016 Atlantic Causal Inference Conference, sought to make progress with respect to both of these issues. The researchers creating the data testing grounds were distinct from the researchers submitting methods whose efﬁcacy would be evaluated. Results from 30 competitors across the two versions of the competition (black-box algorithms and do-it-yourself analyses) are presented along with post-hoc analyses that reveal information about the characteristics of causal inference strategies and settings that affect performance. The most consistent conclusion was that methods that ﬂexibly model the response surface perform better overall than methods that fail to do so. Finally new methods are proposed that combine features of several of the top-performing submitted methods.},
	language = {en},
	number = {1},
	urldate = {2023-06-09},
	journal = {Statistical Science},
	author = {Dorie, Vincent and Hill, Jennifer and Shalit, Uri and Scott, Marc and Cervone, Dan},
	month = feb,
	year = {2019},
	file = {Dorie et al. - 2019 - Automated versus Do-It-Yourself Methods for Causal.pdf:/Users/jacquelinemaasch/Zotero/storage/4MI2U43T/Dorie et al. - 2019 - Automated versus Do-It-Yourself Methods for Causal.pdf:application/pdf},
}

@article{richardson_discussion_2018,
	title = {Discussion of “{Data}-driven confounder selection via {Markov} and {Bayesian} networks” by {Häggström}: {Discussion}},
	volume = {74},
	issn = {0006341X},
	shorttitle = {Discussion of “{Data}-driven confounder selection via {Markov} and {Bayesian} networks” by {Häggström}},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12784},
	doi = {10.1111/biom.12784},
	language = {en},
	number = {2},
	urldate = {2023-06-09},
	journal = {Biometrics},
	author = {Richardson, Thomas S. and Robins, James M. and Wang, Linbo},
	month = jun,
	year = {2018},
	pages = {403--406},
	file = {Richardson et al. - 2018 - Discussion of “Data-driven confounder selection vi.pdf:/Users/jacquelinemaasch/Zotero/storage/MCWHUBTW/Richardson et al. - 2018 - Discussion of “Data-driven confounder selection vi.pdf:application/pdf},
}

@article{saeed_bernoulli_nodate,
	title = {Bernoulli {Mixture} {Models} for {Markov} {Blanket} {Filtering} and {Classiﬁcation}},
	abstract = {This paper presents the use of Bernoulli mixture models for Markov blanket ﬁltering and classiﬁcation of binary data. Bernoulli mixture models can be seen as a tool for partitioning an n-dimensional hypercube, identifying regions of high data density on the corners of the hypercube. Once Bernoulli mixture models are computed from a training dataset we use them for determining the Markov blanket of the target variable. An algorithm for Markov blanket ﬁltering was proposed by Koller and Sahami (1996), which is a greedy search method for feature subset selection and it outputs an approximation to the optimal feature selection criterion. However, they use the entire training instances for computing the conditioning sets and have to limit the size of these sets for computational eﬃciency and avoiding data fragmentation. We have adapted their algorithm to use Bernoulli mixture models instead, hence, overcoming the short comings of their algorithm and increasing the eﬃciency of this algorithm considerably. Once a feature subset is identiﬁed we perform classiﬁcation using these mixture models. We have applied this algorithm to the causality challenge datasets. Our prediction scores were ranked fourth on SIDO and our feature scores were ranked the best for test sets 1 and 2 of the same dataset.},
	language = {en},
	author = {Saeed, Mehreen},
	file = {Saeed - Bernoulli Mixture Models for Markov Blanket Filter.pdf:/Users/jacquelinemaasch/Zotero/storage/MJRH3H9I/Saeed - Bernoulli Mixture Models for Markov Blanket Filter.pdf:application/pdf},
}

@article{kennedy_discussion_2018,
	title = {Discussion of “{Data}-driven confounder selection via {Markov} and {Bayesian} networks” by {Jenny} {Häggström}: {Discussion}},
	volume = {74},
	issn = {0006341X},
	shorttitle = {Discussion of “{Data}-driven confounder selection via {Markov} and {Bayesian} networks” by {Jenny} {Häggström}},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/biom.12787},
	doi = {10.1111/biom.12787},
	language = {en},
	number = {2},
	urldate = {2023-06-09},
	journal = {Biometrics},
	author = {Kennedy, Edward H. and Balakrishnan, Sivaraman},
	month = jun,
	year = {2018},
	pages = {399--402},
	file = {Kennedy and Balakrishnan - 2018 - Discussion of “Data-driven confounder selection vi.pdf:/Users/jacquelinemaasch/Zotero/storage/HJKU9Y9J/Kennedy and Balakrishnan - 2018 - Discussion of “Data-driven confounder selection vi.pdf:application/pdf},
}

@inproceedings{li_data-driven_2018,
	address = {Gothenburg, Sweden},
	title = {{DATA}-{DRIVEN} {RANKING} {AND} {SELECTION}: {HIGH}-{DIMENSIONAL} {COVARIATES} {AND} {GENERAL} {DEPENDENCE}},
	isbn = {978-1-5386-6572-5},
	shorttitle = {{DATA}-{DRIVEN} {RANKING} {AND} {SELECTION}},
	url = {https://ieeexplore.ieee.org/document/8632388/},
	doi = {10.1109/WSC.2018.8632388},
	abstract = {This paper considers the problem of ranking and selection with covariates and aims to identify a decision rule that stipulates the best alternative as a function of the observable covariates. We propose a general data-driven framework to accommodate (i) high-dimensional covariates and (ii) general (nonlinear) dependence between the mean performance of an alternative and the covariates. For both scenarios, we design new selection procedures and provide certain statistical guarantees, by leveraging the data-intensive environment and various statistical learning tools. The performances of our procedures are exhibited through simulation experiments.},
	language = {en},
	urldate = {2023-06-09},
	booktitle = {2018 {Winter} {Simulation} {Conference} ({WSC})},
	publisher = {IEEE},
	author = {Li, Xiaocheng and Zhang, Xiaowei and Zheng, Zeyu},
	month = dec,
	year = {2018},
	pages = {1933--1944},
	file = {Li et al. - 2018 - DATA-DRIVEN RANKING AND SELECTION HIGH-DIMENSIONA.pdf:/Users/jacquelinemaasch/Zotero/storage/4QPIJQGE/Li et al. - 2018 - DATA-DRIVEN RANKING AND SELECTION HIGH-DIMENSIONA.pdf:application/pdf},
}

@article{hunermund_double_2023,
	title = {Double machine learning and automated confounder selection: {A} cautionary tale},
	volume = {11},
	issn = {2193-3685},
	shorttitle = {Double machine learning and automated confounder selection},
	url = {https://www.degruyter.com/document/doi/10.1515/jci-2022-0078/html},
	doi = {10.1515/jci-2022-0078},
	abstract = {Double machine learning (DML) has become an increasingly popular tool for automated variable selection in high-dimensional settings. Even though the ability to deal with a large number of potential covariates can render selection-on-observables assumptions more plausible, there is at the same time a growing risk that endogenous variables are included, which would lead to the violation of conditional independence. This article demonstrates that DML is very sensitive to the inclusion of only a few “bad controls” in the covariate space. The resulting bias varies with the nature of the theoretical causal model, which raises concerns about the feasibility of selecting control variables in a data-driven way.},
	language = {en},
	number = {1},
	urldate = {2023-06-09},
	journal = {Journal of Causal Inference},
	author = {Hünermund, Paul and Louw, Beyers and Caspi, Itamar},
	month = may,
	year = {2023},
	pages = {20220078},
	file = {Hünermund et al. - 2023 - Double machine learning and automated confounder s.pdf:/Users/jacquelinemaasch/Zotero/storage/X5IAEKNV/Hünermund et al. - 2023 - Double machine learning and automated confounder s.pdf:application/pdf},
}

@article{wyss_erratum_2018,
	title = {Erratum: {High}-dimensional {Propensity} {Score} {Adjustment} in {Studies} of {Treatment} {Effects} {Using} {Health} {Care} {Claims} {Data}},
	volume = {29},
	issn = {1044-3983},
	shorttitle = {Erratum},
	url = {https://journals.lww.com/00001648-201811000-00034},
	doi = {10.1097/EDE.0000000000000886},
	language = {en},
	number = {6},
	urldate = {2023-06-09},
	journal = {Epidemiology},
	author = {Wyss, Richard and Fireman, Bruce and Rassen, Jeremy A. and Schneeweiss, Sebastian},
	month = nov,
	year = {2018},
	pages = {e63--e64},
	file = {Wyss et al. - 2018 - Erratum High-dimensional Propensity Score Adjustm.pdf:/Users/jacquelinemaasch/Zotero/storage/QFAQNY8L/Wyss et al. - 2018 - Erratum High-dimensional Propensity Score Adjustm.pdf:application/pdf},
}

@article{schneeweiss_high-dimensional_2009,
	title = {High-dimensional {Propensity} {Score} {Adjustment} in {Studies} of {Treatment} {Effects} {Using} {Health} {Care} {Claims} {Data}},
	volume = {20},
	issn = {1044-3983},
	url = {https://journals.lww.com/00001648-200907000-00008},
	doi = {10.1097/EDE.0b013e3181a663cc},
	abstract = {Background: Adjusting for large numbers of covariates ascertained from patients’ health care claims data may improve control of confounding, as these variables may collectively be proxies for unobserved factors. Here, we develop and test an algorithm that empirically identiﬁes candidate covariates, prioritizes covariates, and integrates them into a propensity-score-based confounder adjustment model.
Methods: We developed a multistep algorithm to implement highdimensional proxy adjustment in claims data. Steps include (1) identifying data dimensions, eg, diagnoses, procedures, and medications; (2) empirically identifying candidate covariates; (3) assessing recurrence of codes; (4) prioritizing covariates; (5) selecting covariates for adjustment; (6) estimating the exposure propensity score; and (7) estimating an outcome model. This algorithm was tested in Medicare claims data, including a study on the effect of Cox-2 inhibitors on reduced gastric toxicity compared with nonselective nonsteroidal anti-inﬂammatory drugs (NSAIDs).
Results: In a population of 49,653 new users of Cox-2 inhibitors or nonselective NSAIDs, a crude relative risk (RR) for upper GI toxicity (RR ϭ 1.09 ͓95\% conﬁdence interval ϭ 0.91–1.30͔) was initially observed. Adjusting for 15 predeﬁned covariates resulted in a possible gastroprotective effect (0.94 ͓0.78 –1.12͔). A gastroprotective effect became stronger when adjusting for an additional 500 algorithm-derived covariates (0.88 ͓0.73–1.06͔). Results of a study on the effect of statin on reduced mortality were similar. Using the algorithm adjustment conﬁrmed a null ﬁnding between inﬂuenza vaccination and hip fracture (1.02 ͓0.85–1.21͔).
Conclusions: In typical pharmacoepidemiologic studies, the proposed high-dimensional propensity score resulted in improved effect estimates compared with adjustment limited to predeﬁned covari-},
	language = {en},
	number = {4},
	urldate = {2023-06-09},
	journal = {Epidemiology},
	author = {Schneeweiss, Sebastian and Rassen, Jeremy A. and Glynn, Robert J. and Avorn, Jerry and Mogun, Helen and Brookhart, M Alan},
	month = jul,
	year = {2009},
	pages = {512--522},
	file = {Schneeweiss et al. - 2009 - High-dimensional Propensity Score Adjustment in St.pdf:/Users/jacquelinemaasch/Zotero/storage/4PPWMXPH/Schneeweiss et al. - 2009 - High-dimensional Propensity Score Adjustment in St.pdf:application/pdf},
}

@article{rassen_covariate_2011,
	title = {Covariate {Selection} in {High}-{Dimensional} {Propensity} {Score} {Analyses} of {Treatment} {Effects} in {Small} {Samples}},
	volume = {173},
	issn = {1476-6256, 0002-9262},
	url = {https://academic.oup.com/aje/article-lookup/doi/10.1093/aje/kwr001},
	doi = {10.1093/aje/kwr001},
	language = {en},
	number = {12},
	urldate = {2023-06-09},
	journal = {American Journal of Epidemiology},
	author = {Rassen, Jeremy A. and Glynn, Robert J. and Brookhart, M. Alan and Schneeweiss, Sebastian},
	month = jun,
	year = {2011},
	pages = {1404--1413},
	file = {Rassen et al. - 2011 - Covariate Selection in High-Dimensional Propensity.pdf:/Users/jacquelinemaasch/Zotero/storage/K2CVS3MD/Rassen et al. - 2011 - Covariate Selection in High-Dimensional Propensity.pdf:application/pdf},
}

@article{franklin_regularized_2015,
	title = {Regularized {Regression} {Versus} the {High}-{Dimensional} {Propensity} {Score} for {Confounding} {Adjustment} in {Secondary} {Database} {Analyses}},
	volume = {182},
	issn = {0002-9262, 1476-6256},
	url = {https://academic.oup.com/aje/article-lookup/doi/10.1093/aje/kwv108},
	doi = {10.1093/aje/kwv108},
	language = {en},
	number = {7},
	urldate = {2023-06-09},
	journal = {American Journal of Epidemiology},
	author = {Franklin, Jessica M. and Eddings, Wesley and Glynn, Robert J. and Schneeweiss, Sebastian},
	month = oct,
	year = {2015},
	pages = {651--659},
	file = {Franklin et al. - 2015 - Regularized Regression Versus the High-Dimensional.pdf:/Users/jacquelinemaasch/Zotero/storage/NG4T2PRQ/Franklin et al. - 2015 - Regularized Regression Versus the High-Dimensional.pdf:application/pdf},
}

@article{karim_can_2018,
	title = {Can {We} {Train} {Machine} {Learning} {Methods} to {Outperform} the {High}-dimensional {Propensity} {Score} {Algorithm}?:},
	volume = {29},
	issn = {1044-3983},
	shorttitle = {Can {We} {Train} {Machine} {Learning} {Methods} to {Outperform} the {High}-dimensional {Propensity} {Score} {Algorithm}?},
	url = {http://journals.lww.com/00001648-201803000-00005},
	doi = {10.1097/EDE.0000000000000787},
	language = {en},
	number = {2},
	urldate = {2023-06-09},
	journal = {Epidemiology},
	author = {Karim, Mohammad Ehsanul and Pang, Menglan and Platt, Robert W.},
	month = mar,
	year = {2018},
	pages = {191--198},
	file = {Karim et al. - 2018 - Can We Train Machine Learning Methods to Outperfor.pdf:/Users/jacquelinemaasch/Zotero/storage/YID387LG/Karim et al. - 2018 - Can We Train Machine Learning Methods to Outperfor.pdf:application/pdf},
}

@article{wyss_using_2018,
	title = {Using {Super} {Learner} {Prediction} {Modeling} to {Improve} {High}-dimensional {Propensity} {Score} {Estimation}:},
	volume = {29},
	issn = {1044-3983},
	shorttitle = {Using {Super} {Learner} {Prediction} {Modeling} to {Improve} {High}-dimensional {Propensity} {Score} {Estimation}},
	url = {http://journals.lww.com/00001648-201801000-00013},
	doi = {10.1097/EDE.0000000000000762},
	abstract = {The high-dimensional propensity score is a semiautomated variable selection algorithm that can supplement expert knowledge to improve confounding control in nonexperimental medical studies utilizing electronic healthcare databases. Although the algorithm can be used to generate hundreds of patient-level variables and rank them by their potential confounding impact, it remains unclear how to select the optimal number of variables for adjustment. We used plasmode simulations based on empirical data to discuss and evaluate data-adaptive approaches for variable selection and prediction modeling that can be combined with the high-dimensional propensity score to improve confounding control in large healthcare databases. We considered approaches that combine the high-dimensional propensity score with Super Learner prediction modeling, a scalable version of collaborative targeted maximum-likelihood estimation, and penalized regression. We evaluated performance using bias and mean squared error (MSE) in effect estimates. Results showed that the high-dimensional propensity score can be sensitive to the number of variables included for adjustment and that severe overfitting of the propensity score model can negatively impact the properties of effect estimates. Combining the high-dimensional propensity score with Super Learner was the most consistent strategy, in terms of reducing bias and MSE in the effect estimates, and may be promising for semiautomated data-adaptive propensity score estimation in highdimensional covariate datasets.},
	language = {en},
	number = {1},
	urldate = {2023-06-09},
	journal = {Epidemiology},
	author = {Wyss, Richard and Schneeweiss, Sebastian and Van Der Laan, Mark and Lendle, Samuel D. and Ju, Cheng and Franklin, Jessica M.},
	month = jan,
	year = {2018},
	pages = {96--106},
	file = {Wyss et al. - 2018 - Using Super Learner Prediction Modeling to Improve.pdf:/Users/jacquelinemaasch/Zotero/storage/VZNA4UMV/Wyss et al. - 2018 - Using Super Learner Prediction Modeling to Improve.pdf:application/pdf},
}

@misc{cheng_data-driven_2022,
	title = {Data-{Driven} {Causal} {Effect} {Estimation} {Based} on {Graphical} {Causal} {Modelling}: {A} {Survey}},
	shorttitle = {Data-{Driven} {Causal} {Effect} {Estimation} {Based} on {Graphical} {Causal} {Modelling}},
	url = {http://arxiv.org/abs/2208.09590},
	abstract = {In many fields of scientific research and real-world applications, unbiased estimation of causal effects from non-experimental data is crucial for understanding the mechanism underlying the data and for decision-making on effective responses or interventions. A great deal of research has been conducted on this challenging problem from different angles. For causal effect estimation in data, assumptions such as Markov property, faithfulness and causal sufficiency are always made. Under the assumptions, full knowledge such as, a set of covariates or an underlying causal graph, is still required. A practical challenge is that in many applications, no such full knowledge or only some partial knowledge is available. In recent years, research has emerged to use a search strategy based on graphical causal modelling to discover useful knowledge from data for causal effect estimation, with some mild assumptions, and has shown promose in tackling the practical challenge. In this survey, we review the methods and focus on the challenges the data-driven methods face. We discuss the assumptions, strengths and limitations of the data-driven methods. We hope this review will motivate more researchers to design better data-driven methods based on graphical causal modelling for the challenging problem of causal effect estimation.},
	language = {en},
	urldate = {2023-06-09},
	publisher = {arXiv},
	author = {Cheng, Debo and Li, Jiuyong and Liu, Lin and Liu, Jixue and Le, Thuc Duy},
	month = aug,
	year = {2022},
	note = {arXiv:2208.09590 [cs]},
	keywords = {Computer Science - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Cheng et al. - 2022 - Data-Driven Causal Effect Estimation Based on Grap.pdf:/Users/jacquelinemaasch/Zotero/storage/W53JAICM/Cheng et al. - 2022 - Data-Driven Causal Effect Estimation Based on Grap.pdf:application/pdf},
}

@article{keller_variable_2020,
	title = {Variable {Selection} for {Causal} {Effect} {Estimation}: {Nonparametric} {Conditional} {Independence} {Testing} {With} {Random} {Forests}},
	volume = {45},
	issn = {1076-9986, 1935-1054},
	shorttitle = {Variable {Selection} for {Causal} {Effect} {Estimation}},
	url = {http://journals.sagepub.com/doi/10.3102/1076998619872001},
	doi = {10.3102/1076998619872001},
	abstract = {Widespread availability of rich educational databases facilitates the use of conditioning strategies to estimate causal effects with nonexperimental data. With dozens, hundreds, or more potential predictors, variable selection can be useful for practical reasons related to communicating results and for statistical reasons related to improving the efficiency of estimators. Background knowledge should take precedence in deciding which variables to retain. However, with many potential predictors, theory may be weak, such that functional form relationships are likely to be unknown. In this article, I propose a nonparametric method for data-driven variable selection based on permutation testing with conditional random forest variable importance. The algorithm automatically handles nonlinear relationships and interactions in its naive implementation. Through a series of Monte Carlo simulation studies and a case study with Early Childhood Longitudinal Study–K data, I find that the method performs well across a variety of scenarios where other methods fail.},
	language = {en},
	number = {2},
	urldate = {2023-06-09},
	journal = {Journal of Educational and Behavioral Statistics},
	author = {Keller, Bryan},
	month = apr,
	year = {2020},
	pages = {119--142},
	file = {Keller - 2020 - Variable Selection for Causal Effect Estimation N.pdf:/Users/jacquelinemaasch/Zotero/storage/W6ZLYAXP/Keller - 2020 - Variable Selection for Causal Effect Estimation N.pdf:application/pdf},
}

@article{sieswerda_identifying_2023,
	title = {Identifying {Confounders} {Using} {Bayesian} {Networks} and {Estimating} {Treatment} {Effect} in {Prostate} {Cancer} {With} {Observational} {Data}},
	issn = {2473-4276},
	url = {https://ascopubs.org/doi/10.1200/CCI.22.00080},
	doi = {10.1200/CCI.22.00080},
	abstract = {PURPOSE Randomized controlled trials are considered the golden standard for estimating treatment effect but are costly to perform and not always possible. Observational data, although readily available, is sensitive to biases such as confounding by indication. Structure learning algorithms for Bayesian Networks (BNs) can be used to discover the underlying model from data. This enables identiﬁcation of confounders through graph analysis, although the model might contain noncausal edges. We propose using a blacklist to aid structure learning in ﬁnding causal relationships. This is illustrated by an analysis into the effect of active treatment (v observation) in localized prostate cancer.
METHODS In total, 4,121 prostate cancer records were obtained from the Netherlands Cancer Registry. Subsequently, we developed a (causal) BN using structure learning while precluding noncausal relations. Additionally, we created several Cox proportional hazards models, each correcting for a different set of potential confounders (including propensity scores). Model predictions for overall survival were compared with expected survival on the basis of the general population using data from Statistics Netherlands (Centraal Bureau voor de Statistiek).
RESULTS Structure learning precluding noncausal relations resulted in a causal graph but did not identify signiﬁcant edges toward treatment; they were added manually. Graph analysis identiﬁed year of diagnosis and age as confounders. The BN predicted a treatment effect of 1 percentage point at 10 years. Chi-squared analysis found signiﬁcant associations between year of diagnosis, age, stage, and treatment. Propensity score correction was successful. Adjusted Cox models predicted signiﬁcant treatment effect around 3 percentage points at 10 years.
CONCLUSION A blacklist in conjunction with structure learning can result in a causal BN that can be used for confounder identiﬁcation. Treatment effect found here is close to the 5 percentage point found in the literature.},
	language = {en},
	number = {7},
	urldate = {2023-06-09},
	journal = {JCO Clinical Cancer Informatics},
	author = {Sieswerda, Melle and Xie, Shixuan and Van Rossum, Ruby and Bermejo, Inigo and Geleijnse, Gijs and Aben, Katja and Van Erning, Felice and Lemmens, Valery and Dekker, André and Verbeek, Xander},
	month = jan,
	year = {2023},
	pages = {e2200080},
	file = {Sieswerda et al. - 2023 - Identifying Confounders Using Bayesian Networks an.pdf:/Users/jacquelinemaasch/Zotero/storage/LW3F3RKA/Sieswerda et al. - 2023 - Identifying Confounders Using Bayesian Networks an.pdf:application/pdf},
}

@misc{gao_efficient_2015,
	title = {Efficient {Estimation} of {Mutual} {Information} for {Strongly} {Dependent} {Variables}},
	url = {http://arxiv.org/abs/1411.2003},
	abstract = {We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large sample size. This important yet overlooked shortcoming of the existing estimators is due to their implicit reliance on local uniformity of the underlying joint distribution. We introduce a new estimator that is robust to local non-uniformity, works well with limited data, and is able to capture relationship strengths over many orders of magnitude. We demonstrate the superior performance of the proposed estimator on both synthetic and real-world data.},
	language = {en},
	urldate = {2023-06-16},
	publisher = {arXiv},
	author = {Gao, Shuyang and Steeg, Greg Ver and Galstyan, Aram},
	month = mar,
	year = {2015},
	note = {arXiv:1411.2003 [physics, stat]},
	keywords = {Statistics - Machine Learning, Computer Science - Information Theory, Physics - Data Analysis, Statistics and Probability},
	file = {Gao et al. - 2015 - Efficient Estimation of Mutual Information for Str.pdf:/Users/jacquelinemaasch/Zotero/storage/H3SPTHED/Gao et al. - 2015 - Efficient Estimation of Mutual Information for Str.pdf:application/pdf},
}

@article{liu_evaluating_2021,
	title = {Evaluating eligibility criteria of oncology trials using real-world data and {AI}},
	volume = {592},
	issn = {0028-0836, 1476-4687},
	url = {https://www.nature.com/articles/s41586-021-03430-5},
	doi = {10.1038/s41586-021-03430-5},
	language = {en},
	number = {7855},
	urldate = {2023-07-05},
	journal = {Nature},
	author = {Liu, Ruishan and Rizzo, Shemra and Whipple, Samuel and Pal, Navdeep and Pineda, Arturo Lopez and Lu, Michael and Arnieri, Brandon and Lu, Ying and Capra, William and Copping, Ryan and Zou, James},
	month = apr,
	year = {2021},
	pages = {629--633},
	file = {Liu et al. - 2021 - Evaluating eligibility criteria of oncology trials.pdf:/Users/jacquelinemaasch/Zotero/storage/ATNC9TRN/Liu et al. - 2021 - Evaluating eligibility criteria of oncology trials.pdf:application/pdf},
}

@inproceedings{chien_multi-disciplinary_2022,
	address = {Seoul Republic of Korea},
	title = {Multi-disciplinary fairness considerations in machine learning for clinical trials},
	isbn = {978-1-4503-9352-2},
	url = {https://dl.acm.org/doi/10.1145/3531146.3533154},
	doi = {10.1145/3531146.3533154},
	abstract = {While interest in the application of machine learning to improve healthcare has grown tremendously in recent years, a number of barriers prevent deployment in medical practice. A notable concern is the potential to exacerbate entrenched biases and existing health disparities in society. The area of fairness in machine learning seeks to address these issues of equity; however, appropriate approaches are context-dependent, necessitating domain-specific consideration. We focus on clinical trials, i.e., research studies conducted on humans to evaluate medical treatments. Clinical trials are a relatively under-explored application in machine learning for healthcare, in part due to complex ethical, legal, and regulatory requirements and high costs. Our aim is to provide a multi-disciplinary assessment of how fairness for machine learning fits into the context of clinical trials research and practice. We start by reviewing the current ethical considerations and guidelines for clinical trials and examine their relationship with common definitions of fairness in machine learning. We examine potential sources of unfairness in clinical trials, providing concrete examples, and discuss the role machine learning might play in either mitigating potential biases or exacerbating them when applied without care. Particular focus is given to adaptive clinical trials, which may employ machine learning. Finally, we highlight concepts that require further investigation and development, and emphasize new approaches to fairness that may be relevant to the design of clinical trials.},
	language = {en},
	urldate = {2023-07-05},
	booktitle = {2022 {ACM} {Conference} on {Fairness}, {Accountability}, and {Transparency}},
	publisher = {ACM},
	author = {Chien, Isabel and Deliu, Nina and Turner, Richard and Weller, Adrian and Villar, Sofia and Kilbertus, Niki},
	month = jun,
	year = {2022},
	pages = {906--924},
	file = {Chien et al. - 2022 - Multi-disciplinary fairness considerations in mach.pdf:/Users/jacquelinemaasch/Zotero/storage/WBXYS5KQ/Chien et al. - 2022 - Multi-disciplinary fairness considerations in mach.pdf:application/pdf},
}

@article{tan_digital_2022,
	title = {Digital approaches to enhancing community engagement in clinical trials},
	volume = {5},
	issn = {2398-6352},
	url = {https://www.nature.com/articles/s41746-022-00581-1},
	doi = {10.1038/s41746-022-00581-1},
	abstract = {Abstract
            Digital approaches are increasingly common in clinical trial recruitment, retention, analysis, and dissemination. Community engagement processes have contributed to the successful implementation of clinical trials and are crucial in enhancing equity in trials. However, few studies focus on how digital approaches can be implemented to enhance community engagement in clinical trials. This narrative review examines three key areas for digital approaches to deepen community engagement in clinical trials—the use of digital technology for trial processes to decentralize trials, digital crowdsourcing to develop trial components, and digital qualitative research methods. We highlight how digital approaches enhanced community engagement through a greater diversity of participants, and deepened community engagement through the decentralization of research processes. We discuss new possibilities that digital technologies offer for community engagement, and highlight potential strengths, weaknesses, and practical considerations. We argue that strengthening community engagement using a digital approach can enhance equity and improve health outcomes.},
	language = {en},
	number = {1},
	urldate = {2023-07-05},
	journal = {npj Digital Medicine},
	author = {Tan, Rayner K. J. and Wu, Dan and Day, Suzanne and Zhao, Yang and Larson, Heidi J. and Sylvia, Sean and Tang, Weiming and Tucker, Joseph D.},
	month = mar,
	year = {2022},
	pages = {37},
	file = {Tan et al. - 2022 - Digital approaches to enhancing community engageme.pdf:/Users/jacquelinemaasch/Zotero/storage/TYTCTTQ8/Tan et al. - 2022 - Digital approaches to enhancing community engageme.pdf:application/pdf},
}

@article{inan_digitizing_2020,
	title = {Digitizing clinical trials},
	volume = {3},
	issn = {2398-6352},
	url = {https://www.nature.com/articles/s41746-020-0302-y},
	doi = {10.1038/s41746-020-0302-y},
	abstract = {Abstract
            Clinical trials are a fundamental tool used to evaluate the efficacy and safety of new drugs and medical devices and other health system interventions. The traditional clinical trials system acts as a quality funnel for the development and implementation of new drugs, devices and health system interventions. The concept of a “digital clinical trial” involves leveraging digital technology to improve participant access, engagement, trial-related measurements, and/or interventions, enable concealed randomized intervention allocation, and has the potential to transform clinical trials and to lower their cost. In April 2019, the US National Institutes of Health (NIH) and the National Science Foundation (NSF) held a workshop bringing together experts in clinical trials, digital technology, and digital analytics to discuss strategies to implement the use of digital technologies in clinical trials while considering potential challenges. This position paper builds on this workshop to describe the current state of the art for digital clinical trials including (1) defining and outlining the composition and elements of digital trials; (2) describing recruitment and retention using digital technology; (3) outlining data collection elements including mobile health, wearable technologies, application programming interfaces (APIs), digital transmission of data, and consideration of regulatory oversight and guidance for data security, privacy, and remotely provided informed consent; (4) elucidating digital analytics and data science approaches leveraging artificial intelligence and machine learning algorithms; and (5) setting future priorities and strategies that should be addressed to successfully harness digital methods and the myriad benefits of such technologies for clinical research.},
	language = {en},
	number = {1},
	urldate = {2023-07-05},
	journal = {npj Digital Medicine},
	author = {Inan, O. T. and Tenaerts, P. and Prindiville, S. A. and Reynolds, H. R. and Dizon, D. S. and Cooper-Arnold, K. and Turakhia, M. and Pletcher, M. J. and Preston, K. L. and Krumholz, H. M. and Marlin, B. M. and Mandl, K. D. and Klasnja, P. and Spring, B. and Iturriaga, E. and Campo, R. and Desvigne-Nickens, P. and Rosenberg, Y. and Steinhubl, S. R. and Califf, R. M.},
	month = jul,
	year = {2020},
	pages = {101},
	file = {Inan et al. - 2020 - Digitizing clinical trials.pdf:/Users/jacquelinemaasch/Zotero/storage/WANDLVS3/Inan et al. - 2020 - Digitizing clinical trials.pdf:application/pdf},
}

@article{steinhubl_digital_2019,
	title = {Digital clinical trials: creating a vision for the future},
	volume = {2},
	issn = {2398-6352},
	shorttitle = {Digital clinical trials},
	url = {https://www.nature.com/articles/s41746-019-0203-0},
	doi = {10.1038/s41746-019-0203-0},
	language = {en},
	number = {1},
	urldate = {2023-07-05},
	journal = {npj Digital Medicine},
	author = {Steinhubl, Steven R. and Wolff-Hughes, Dana L. and Nilsen, Wendy and Iturriaga, Erin and Califf, Robert M.},
	month = dec,
	year = {2019},
	pages = {126, s41746--019--0203--0},
	file = {Steinhubl et al. - 2019 - Digital clinical trials creating a vision for the.pdf:/Users/jacquelinemaasch/Zotero/storage/428KHATW/Steinhubl et al. - 2019 - Digital clinical trials creating a vision for the.pdf:application/pdf},
}

@article{labrecque_target_2017,
	title = {Target trial emulation: teaching epidemiology and beyond},
	volume = {32},
	issn = {0393-2990, 1573-7284},
	shorttitle = {Target trial emulation},
	url = {http://link.springer.com/10.1007/s10654-017-0293-4},
	doi = {10.1007/s10654-017-0293-4},
	language = {en},
	number = {6},
	urldate = {2023-07-05},
	journal = {European Journal of Epidemiology},
	author = {Labrecque, Jeremy A. and Swanson, Sonja A.},
	month = jun,
	year = {2017},
	pages = {473--475},
	file = {Labrecque and Swanson - 2017 - Target trial emulation teaching epidemiology and .pdf:/Users/jacquelinemaasch/Zotero/storage/CE6S7EES/Labrecque and Swanson - 2017 - Target trial emulation teaching epidemiology and .pdf:application/pdf},
}

@article{woo_trial_2019,
	title = {Trial by artificial intelligence},
	volume = {573},
	language = {en},
	journal = {Nature},
	author = {Woo, Marcus},
	year = {2019},
	pages = {S100--S102},
	file = {Woo - Trial by artificial intelligence.pdf:/Users/jacquelinemaasch/Zotero/storage/UT6QUKWX/Woo - Trial by artificial intelligence.pdf:application/pdf},
}

@article{laneman_distribution_nodate,
	title = {On the {Distribution} of {Mutual} {Information}},
	abstract = {In the early years of information theory, mutual information was deﬁned as a random variable, and error probability bounds for communication systems were obtained in terms of its probability distribution. We advocate a return to this perspective for a renewed look at information theory for general channel models and ﬁnite coding blocklengths. For capacityachieving inputs, we characterize the mutual information random variables for several important channel models, including the discrete memoryless binary symmetric channel (BSC), the discrete-time complex additive white Gaussian noise (AWGN) channel, and the discrete-time Rayleigh fading channel with static, ﬂat fading known perfectly to the decoder. Combined with a result known as Feinstein’s lemma and its extensions, these characterizations can be employed to obtain bounds on the maximal block error probability of channel codes operating over these channels. Such bounds appear to be particularly useful when operating at transmission rates near the average mutual information.},
	language = {en},
	author = {Laneman, J Nicholas},
	file = {Laneman - On the Distribution of Mutual Information.pdf:/Users/jacquelinemaasch/Zotero/storage/9U8MLF2E/Laneman - On the Distribution of Mutual Information.pdf:application/pdf},
}

@article{hutter_distribution_nodate,
	title = {Distribution of {Mutual} {Information}},
	abstract = {The mutual information of two random variables z and J with joint probabilities \{7rij\} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point estimate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(7r) comprising prior information about 7r. From the prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(Iln). We concentrate on the mean, variance, skewness, and kurtosis , and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.},
	language = {en},
	author = {Hutter, Marcus},
	file = {Hutter - Distribution of Mutual Information.pdf:/Users/jacquelinemaasch/Zotero/storage/FHKDR7UZ/Hutter - Distribution of Mutual Information.pdf:application/pdf},
}

@inproceedings{goebel_approximation_2005,
	address = {Seoul, Korea},
	title = {An approximation to the distribution of finite sample size mutual information estimates},
	volume = {2},
	isbn = {978-0-7803-8938-0},
	url = {http://ieeexplore.ieee.org/document/1494518/},
	doi = {10.1109/ICC.2005.1494518},
	abstract = {In this paper, the distribution of mutual information between two discrete random variables is approximated by means of a second-order Taylor series expansion. Approximative expressions for the distribution of mutual information (MI) between independent random variables, conditional MI between conditionally independent variables, and MI between (weakly) dependent random variables are derived. These distributions are functions of the available sample size and the number of realisations of the random variables only; knowledge of the variables’ PMF is not required. The results are veriﬁed numerically for various cases. Exemplary application ideas in statistics and communications engineering are proposed.},
	language = {en},
	urldate = {2023-07-07},
	booktitle = {{IEEE} {International} {Conference} on {Communications}, 2005. {ICC} 2005. 2005},
	publisher = {IEEE},
	author = {Goebel, B. and Dawy, Z. and Hagenauer, J. and Mueller, J.C.},
	year = {2005},
	pages = {1102--1106},
	file = {Goebel et al. - 2005 - An approximation to the distribution of finite sam.pdf:/Users/jacquelinemaasch/Zotero/storage/PJZYGJ95/Goebel et al. - 2005 - An approximation to the distribution of finite sam.pdf:application/pdf},
}

@article{knuth_optimal_2019,
	title = {Optimal data-based binning for histograms and histogram-based probability density models},
	volume = {95},
	issn = {10512004},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S1051200419301277},
	doi = {10.1016/j.dsp.2019.102581},
	language = {en},
	urldate = {2023-07-07},
	journal = {Digital Signal Processing},
	author = {Knuth, Kevin H.},
	month = dec,
	year = {2019},
	pages = {102581},
	file = {Knuth - 2019 - Optimal data-based binning for histograms and hist.pdf:/Users/jacquelinemaasch/Zotero/storage/XCBHENE4/Knuth - 2019 - Optimal data-based binning for histograms and hist.pdf:application/pdf},
}

@article{kleiter_posterior_1999,
	title = {The posterior probability of {Bayes} nets with strong dependences},
	volume = {3},
	issn = {1432-7643, 1433-7479},
	url = {http://link.springer.com/10.1007/s005000050065},
	doi = {10.1007/s005000050065},
	abstract = {Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real-world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map signiﬁcant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. The arc probabilities are set equal to the probability that the mutual information, provided by the neighbors of a node, exceeds a given threshold. A 2 approximation for the probability density function of mutual information is used. A large number of network structures in which the arc probabilities are evaluated, is generated by stochastic simulation. Finally, the probabilities of the whole graph structures are obtained by combining the individual arc probabilities with the number of possible construction sequences compatible with the partial ordering of the graph. While selecting models with large deviance from independence results in simple networks with few but important links, selecting models with small deviance results in highly connected networks also containing less important links.},
	language = {en},
	number = {3},
	urldate = {2023-07-07},
	journal = {Soft Computing - A Fusion of Foundations, Methodologies and Applications},
	author = {Kleiter, G. D.},
	month = nov,
	year = {1999},
	pages = {162--173},
	file = {Kleiter - 1999 - The posterior probability of Bayes nets with stron.pdf:/Users/jacquelinemaasch/Zotero/storage/GGEZDNRX/Kleiter - 1999 - The posterior probability of Bayes nets with stron.pdf:application/pdf},
}

@inproceedings{kleiter_learning_1996,
	title = {Learning {Bayesian} {Networks} under the {Control} of {Mutual} {Information}},
	abstract = {The extraction of the structure of a Bayesian network from data is conceived as a stochastic process of learning a random acyclic directed graph. The nodes of the graph are taken as xed but its arcs are randomly switched on or o . The probabilities for the arcs being in the on or o state are controlled by the amount and the precision of the mutual information the arcs transmit to the neighboring nodes. A 2 approximation for the second order probability density function of mutual information in contingency tables with Dirichlet distributed rst order probabilities is proposed. EM is used for partially observed data. The neighborhood of each node is de ned by its Markov shield. The density function is employed to select only those dependencies for which the probability that the mutual information exceeds a selected minimum mutual information (MMI) criterion is large. The criterion is combined with Gibbs sampling to learn the structure of a Bayesian network from complete or incomplete data. The method allows to nd the posterior probability distribution of Bayesian network structures.},
	language = {en},
	booktitle = {Proceedings of the 6th {International} {Conference} on {Information} {Processing} and {Management} of {Uncertainty} in {Knowledge}-{Based} {Systems} ({IPMU}-1996)},
	author = {Kleiter, Gernot D and Jirousek, Radim},
	year = {1996},
	file = {Kleiter and Jirousek - Learning Bayesian Networks under the Control of Mu.pdf:/Users/jacquelinemaasch/Zotero/storage/SWZ8LZGG/Kleiter and Jirousek - Learning Bayesian Networks under the Control of Mu.pdf:application/pdf},
}

@misc{vollmer_iterative_2019,
	title = {Iterative {Estimation} of {Mutual} {Information} with {Error} {Bounds}},
	url = {https://openproceedings.org/2019/conf/edbt/EDBT19_paper_94.pdf},
	doi = {10.5441/002/EDBT.2019.08},
	abstract = {Mutual Information (MI) is an established measure for linear and nonlinear dependencies between two variables. Estimating MI is nontrivial and requires notable computation power for high estimation quality. While some estimation techniques allow trading result quality for lower runtimes, this tradeoff is fixed per task and cannot be adjusted. If the available time is unknown in advance or is overestimated, one may need to abort the estimation without any result. Conversely, when there are several estimation tasks, and one wants to budget computation time between them, there currently is no efficient way to adjust it dynamically based on certain targets, e.g., high MI values or MI values close to a constant. In this article, we present an iterative estimator of MI. Our method offers an estimate with low quality near-instantly and improves this estimate in fine grained steps with more computation time. The estimate also converges towards the result of a conventional estimator. We prove that the time complexity for this convergence is only slightly slower than non-iterative estimation. Additionally, with each step our estimator also tightens statistical guarantees regarding the convergence result, i.e., confidence intervals, progressively. These also serve as quality indicators for early estimates and allow to reliably discern between attribute pairs with weak and strong dependencies. Our experiments show that these guarantees can also be used to execute threshold queries faster compared to non-iterative estimation.},
	language = {en},
	urldate = {2023-07-07},
	publisher = {OpenProceedings.org},
	author = {Vollmer, Michael and Böhm, Klemens},
	year = {2019},
	keywords = {Database Technology},
	file = {Vollmer and Böhm - 2019 - Iterative Estimation of Mutual Information with Er.pdf:/Users/jacquelinemaasch/Zotero/storage/TRS4ZNXT/Vollmer and Böhm - 2019 - Iterative Estimation of Mutual Information with Er.pdf:application/pdf},
}

@article{singh_finite-sample_nodate,
	title = {Finite-{Sample} {Analysis} of {Fixed}-k {Nearest} {Neighbor} {Density} {Functional} {Estimators}},
	abstract = {We provide ﬁnite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability density, including entropies and divergences. Rather than plugging a consistent density estimate (which requires k → ∞ as the sample size n → ∞) into the functional of interest, the estimators we consider ﬁx k and perform a bias correction. This is more efﬁcient computationally, and, as we show in certain cases, statistically, leading to faster convergence rates. Our framework uniﬁes several previous estimators, for most of which ours are the ﬁrst ﬁnite sample guarantees.},
	language = {en},
	author = {Singh, Shashank and Poczos, Barnabas},
	file = {Singh and Poczos - Finite-Sample Analysis of Fixed-k Nearest Neighbor.pdf:/Users/jacquelinemaasch/Zotero/storage/AFCFSDIG/Singh and Poczos - Finite-Sample Analysis of Fixed-k Nearest Neighbor.pdf:application/pdf},
}

@article{cataron_asymptotically_2013,
	title = {Asymptotically {Unbiased} {Estimator} of the {Informational} {Energy} with {kNN}},
	volume = {8},
	issn = {1841-9844, 1841-9836},
	url = {http://univagora.ro/jour/index.php/ijccc/article/view/643},
	doi = {10.15837/ijccc.2013.5.643},
	abstract = {Motivated by machine learning applications (e.g., classiﬁcation, function approximation, feature extraction), in previous work, we have introduced a nonparametric estimator of Onicescu’s informational energy. Our method was based on the k-th nearest neighbor distances between the n sample points, where k is a ﬁxed positive integer. In the present contribution, we discuss mathematical properties of this estimator. We show that our estimator is asymptotically unbiased and consistent. We provide further experimental results which illustrate the convergence of the estimator for standard distributions.},
	language = {en},
	number = {5},
	urldate = {2023-07-07},
	journal = {International Journal of Computers Communications \& Control},
	author = {Caţaron, Angel and Andonie, Răzvan and Chueh, Yvonne},
	month = sep,
	year = {2013},
	pages = {689},
	file = {Caţaron et al. - 2013 - Asymptotically Unbiased Estimator of the Informati.pdf:/Users/jacquelinemaasch/Zotero/storage/VFXXPBRC/Caţaron et al. - 2013 - Asymptotically Unbiased Estimator of the Informati.pdf:application/pdf},
}

@article{cheng_learning_2002,
	title = {Learning {Bayesian} networks from data: {An} information-theory based approach},
	volume = {137},
	issn = {00043702},
	shorttitle = {Learning {Bayesian} networks from data},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0004370202001911},
	doi = {10.1016/S0004-3702(02)00191-1},
	abstract = {This paper provides algorithms that use an information-theoretic analysis to learn Bayesian network structures from data. Based on our three-phase learning framework, we develop efﬁcient algorithms that can effectively learn Bayesian networks, requiring only polynomial numbers of conditional independence (CI) tests in typical cases. We provide precise conditions that specify when these algorithms are guaranteed to be correct as well as empirical evidence (from real world applications and simulation tests) that demonstrates that these systems work efﬁciently and reliably in practice.  2002 Elsevier Science B.V. All rights reserved.},
	language = {en},
	number = {1-2},
	urldate = {2023-07-07},
	journal = {Artificial Intelligence},
	author = {Cheng, Jie and Greiner, Russell and Kelly, Jonathan and Bell, David and Liu, Weiru},
	month = may,
	year = {2002},
	pages = {43--90},
	file = {Cheng et al. - 2002 - Learning Bayesian networks from data An informati.pdf:/Users/jacquelinemaasch/Zotero/storage/BB48JPXA/Cheng et al. - 2002 - Learning Bayesian networks from data An informati.pdf:application/pdf},
}

@article{fung_1990-constructor_nodate,
	title = {1990-{Constructor}: {A} {System} for the {Induction} of {Probabilistic} {Models}},
	abstract = {The probabilistic network technology is a knowledgebased technique which focuses on reasoning under uncertainty. Because of its well defined semantics and solid theoretical foundations, the technology is finding increasing application in fields such as medical diagnosis, machine vision, military situation assessment , petroleum exploration, and information retrieval. However, like other knowledge-based techniques, acquiring the qualitative and quantitative information needed to build these networks can be highly labor-intensive.},
	language = {en},
	author = {Fung, Robert M and Crawford, Stuart L},
	file = {Fung and Crawford - 1990-Constructor A System for the Induction of Pr.pdf:/Users/jacquelinemaasch/Zotero/storage/LKCZIGFU/Fung and Crawford - 1990-Constructor A System for the Induction of Pr.pdf:application/pdf},
}

@article{dash_hybrid_nodate,
	title = {A {Hybrid} {Anytime} {Algorithm} for the {Construction} of {Causal} {Models} {From} {Sparse} {Data}},
	abstract = {We present a hybrid constraintbased/Bayesian algorithm for learning causal networks in the presence of sparse data. The algorithm searches the space of equivalence classes of models (essential graphs) using a heuristic based on conventional constraintbased techniques. Each essential graph is then converted into a directed acyclic graph and scored using a Bayesian scoring metric. Two variants of the algorithm are developed and tested using data from randomly generated networks of sizes from 15 to 45 nodes with data sizes ranging from 250 to 2000 records. Both variations are compared to, and found to consistently outperform two variations of greedy search with restarts.},
	language = {en},
	author = {Dash, Denver and Druzdzel, Marek J},
	file = {Dash and Druzdzel - A Hybrid Anytime Algorithm for the Construction of.pdf:/Users/jacquelinemaasch/Zotero/storage/HZTZT7VE/Dash and Druzdzel - A Hybrid Anytime Algorithm for the Construction of.pdf:application/pdf},
}

@article{sober_principle_nodate,
	title = {{THE} {PRINCIPLE} {OF} {THE} {COMMON} {CAUSE}},
	language = {en},
	journal = {THE PRINCIPLE OF THE COMMON CAUSE},
	author = {Sober, Elliott},
	file = {Sober - THE PRINCIPLE OF THE COMMON CAUSE.pdf:/Users/jacquelinemaasch/Zotero/storage/CT2C7Y32/Sober - THE PRINCIPLE OF THE COMMON CAUSE.pdf:application/pdf},
}

@article{hofer-szabo_reichenbachs_1999,
	title = {On {Reichenbach}'s {Common} {Cause} {Principle} and {Reichenbach}'s {Notion} of {Common} {Cause}},
	volume = {50},
	issn = {0007-0882, 1464-3537},
	url = {https://www.journals.uchicago.edu/doi/10.1093/bjps/50.3.377},
	doi = {10.1093/bjps/50.3.377},
	abstract = {It is shown that, given any finite set of pairs of random events in a Boolea which are correlated with respect to a fixed probability measure on the alg algebra can be extended in such a way that the extension contains events th regarded as common causes of the correlations in the sense of Reichenbach's of common cause. It is shown, further, that, given any quantum probability any set of commuting events in it which are correlated with respect to a fixed state, the quantum probability space can be extended in such a way that the contains common causes of all the selected correlations, where common caus taken in the sense of Reichenbach's definition. It is argued that these re strongly restrict the possible ways of disproving Reichenbach's Comm Principle.},
	language = {en},
	number = {3},
	urldate = {2023-07-16},
	journal = {The British Journal for the Philosophy of Science},
	author = {Hofer-Szabó, Gábor and Rédei, Miklós and Szabó, László E.},
	month = sep,
	year = {1999},
	pages = {377--399},
	file = {Hofer-Szabó et al. - 1999 - On Reichenbach's Common Cause Principle and Reiche.pdf:/Users/jacquelinemaasch/Zotero/storage/2FWMYS7F/Hofer-Szabó et al. - 1999 - On Reichenbach's Common Cause Principle and Reiche.pdf:application/pdf},
}

@article{pearl_myth_2000,
	title = {Myth, {Confusion}, and {Science} in {Causal} {Analysis}},
	abstract = {This letter argues that the practice of conditioning on all observed covariates, recently advocated by several analysts, should be treated with great caution. Graphical methods explain why, and provide the scientiﬁc basis for principled selection of covariates.},
	language = {en},
	author = {Pearl, J},
	year = {2000},
	file = {Pearl - 2000 - Myth, Confusion, and Science in Causal Analysis.pdf:/Users/jacquelinemaasch/Zotero/storage/D7JFTGMW/Pearl - 2000 - Myth, Confusion, and Science in Causal Analysis.pdf:application/pdf},
}

@article{shiba_using_2021,
	title = {Using {Propensity} {Scores} for {Causal} {Inference}: {Pitfalls} and {Tips}},
	volume = {31},
	issn = {0917-5040, 1349-9092},
	shorttitle = {Using {Propensity} {Scores} for {Causal} {Inference}},
	url = {https://www.jstage.jst.go.jp/article/jea/31/8/31_JE20210145/_article},
	doi = {10.2188/jea.JE20210145},
	abstract = {Methods based on propensity score (PS) have become increasingly popular as a tool for causal inference. A better understanding of the relative advantages and disadvantages of the alternative analytic approaches can contribute to the optimal choice and use of a speciﬁc PS method over other methods. In this article, we provide an accessible overview of causal inference from observational data and two major PS-based methods (matching and inverse probability weighting), focusing on the underlying assumptions and decision-making processes. We then discuss common pitfalls and tips for applying the PS methods to empirical research and compare the conventional multivariable outcome regression and the two alternative PS-based methods (ie, matching and inverse probability weighting) and discuss their similarities and diﬀerences. Although we note subtle diﬀerences in causal identiﬁcation assumptions, we highlight that the methods are distinct primarily in terms of the statistical modeling assumptions involved and the target population for which exposure eﬀects are being estimated.},
	language = {en},
	number = {8},
	urldate = {2023-07-16},
	journal = {Journal of Epidemiology},
	author = {Shiba, Koichiro and Kawahara, Takuya},
	month = aug,
	year = {2021},
	pages = {457--463},
	file = {Shiba and Kawahara - 2021 - Using Propensity Scores for Causal Inference Pitf.pdf:/Users/jacquelinemaasch/Zotero/storage/XWXC4I48/Shiba and Kawahara - 2021 - Using Propensity Scores for Causal Inference Pitf.pdf:application/pdf},
}

@article{ramsey_million_2017,
	title = {A million variables and more: the {Fast} {Greedy} {Equivalence} {Search} algorithm for learning high-dimensional graphical causal models, with an application to functional magnetic resonance images},
	volume = {3},
	issn = {2364-415X, 2364-4168},
	shorttitle = {A million variables and more},
	url = {http://link.springer.com/10.1007/s41060-016-0032-z},
	doi = {10.1007/s41060-016-0032-z},
	abstract = {We describe two modiﬁcations that parallelize and reorganize caching in the well-known Greedy Equivalence Search algorithm for discovering directed acyclic graphs on random variables from sample values. We apply one of these modiﬁcations, the Fast Greedy Equivalence Search (fGES) assuming faithfulness, to an i.i.d. sample of 1000 units to recover with high precision and good recall an average degree 2 directed acyclic graph with one million Gaussian variables. We describe a modiﬁcation of the algorithm to rapidly ﬁnd the Markov Blanket of any variable in a high dimensional system. Using 51,000 voxels that parcellate an entire human cortex, we apply the fGES algorithm to blood oxygenation level-dependent time series obtained from resting state fMRI.},
	language = {en},
	number = {2},
	urldate = {2023-07-16},
	journal = {International Journal of Data Science and Analytics},
	author = {Ramsey, Joseph and Glymour, Madelyn and Sanchez-Romero, Ruben and Glymour, Clark},
	month = mar,
	year = {2017},
	pages = {121--129},
	file = {Ramsey et al. - 2017 - A million variables and more the Fast Greedy Equi.pdf:/Users/jacquelinemaasch/Zotero/storage/GFQAVX4F/Ramsey et al. - 2017 - A million variables and more the Fast Greedy Equi.pdf:application/pdf},
}

@article{chickering_optimal_2002,
	title = {Optimal {Structure} {Identiﬁcation} {With} {Greedy} {Search}},
	volume = {3},
	abstract = {In this paper we prove the so-called “Meek Conjecture”. In particular, we show that if a DAG H is an independence map of another DAG G, then there exists a ﬁnite sequence of edge additions and covered edge reversals in G such that (1) after each edge modiﬁcation H remains an independence map of G and (2) after all modiﬁcations G = H. As shown by Meek (1997), this result has an important consequence for Bayesian approaches to learning Bayesian networks from data: in the limit of large sample size, there exists a twophase greedy search algorithm that—when applied to a particular sparsely-connected search space—provably identiﬁes a perfect map of the generative distribution if that perfect map is a DAG. We provide a new implementation of the search space, using equivalence classes as states, for which all operators used in the greedy search can be scored eﬃciently using local functions of the nodes in the domain. Finally, using both synthetic and real-world datasets, we demonstrate that the two-phase greedy approach leads to good solutions when learning with ﬁnite sample sizes.},
	language = {en},
	journal = {Journal of Machine Learning Research},
	author = {Chickering, David Maxwell},
	year = {2002},
	file = {Chickering - Optimal Structure Identiﬁcation With Greedy Search.pdf:/Users/jacquelinemaasch/Zotero/storage/KC9H5GNT/Chickering - Optimal Structure Identiﬁcation With Greedy Search.pdf:application/pdf},
}

@article{zhang_estimation_nodate,
	title = {On {Estimation} of {Functional} {Causal} {Models}: {General} {Results} and {Application} to {Post}-{Nonlinear} {Causal} {Model}},
	language = {en},
	journal = {ACM Transactions on Intelligent Systems and Technology},
	author = {Zhang, Kun and Wang, Zhikun and Zhang, Jiji and Schökopf, Bernhard},
	file = {Zhang et al. - On Estimation of Functional Causal Models General.pdf:/Users/jacquelinemaasch/Zotero/storage/MUY2FBSX/Zhang et al. - On Estimation of Functional Causal Models General.pdf:application/pdf},
}

@article{kozachenko_sample_1987,
	title = {Sample {Estimate} of the {Entropy} of a {Random} {Vector}},
	volume = {23},
	language = {ru},
	number = {2},
	journal = {Problemy Peredachi Informatsii},
	author = {Kozachenko, L. F. and Leonenko, N. N},
	year = {1987},
	pages = {9--16},
	file = {Kozachenko and Leonenko - 1987 - Sample Estimate of the Entropy of a Random Vector.pdf:/Users/jacquelinemaasch/Zotero/storage/FG496L3D/Kozachenko and Leonenko - 1987 - Sample Estimate of the Entropy of a Random Vector.pdf:application/pdf},
}

@article{vejmelka_inferring_2008,
	title = {Inferring the directionality of coupling with conditional mutual information},
	volume = {77},
	issn = {1539-3755, 1550-2376},
	url = {https://link.aps.org/doi/10.1103/PhysRevE.77.026214},
	doi = {10.1103/PhysRevE.77.026214},
	language = {en},
	number = {2},
	urldate = {2023-07-27},
	journal = {Physical Review E},
	author = {Vejmelka, Martin and Paluš, Milan},
	month = feb,
	year = {2008},
	pages = {026214},
	file = {Vejmelka and Paluš - 2008 - Inferring the directionality of coupling with cond.pdf:/Users/jacquelinemaasch/Zotero/storage/6M2WHX2B/Vejmelka and Paluš - 2008 - Inferring the directionality of coupling with cond.pdf:application/pdf},
}

@article{benson_comparison_2000,
	title = {A {Comparison} of {Observational} {Studies} and {Randomized}, {Controlled} {Trials}},
	abstract = {Background For many years it has been claimed that observational studies find stronger treatment effects than randomized, controlled trials. We compared the results of observational studies with those of randomized, controlled trials.
Methods We searched the Abridged Index Medicus and Cochrane data bases to identify observational studies reported between 1985 and 1998 that compared two or more treatments or interventions for the same condition. We then searched the Medline and Cochrane data bases to identify all the randomized, controlled trials and observational studies comparing the same treatments for these conditions. For each treatment, the magnitudes of the effects in the various observational studies were combined by the Mantel–Haenszel or weighted analysis-of-variance procedure and then compared with the combined magnitude of the effects in the randomized, controlled trials that evaluated the same treatment.
Results There were 136 reports about 19 diverse treatments, such as calcium-channel–blocker therapy for coronary artery disease, appendectomy, and interventions for subfertility. In most cases, the estimates of the treatment effects from observational studies and randomized, controlled trials were similar. In only 2 of the 19 analyses of treatment effects did the combined magnitude of the effect in observational studies lie outside the 95 percent confidence interval for the combined magnitude in the randomized, controlled trials.
Conclusions We found little evidence that estimates of treatment effects in observational studies reported after 1984 are either consistently larger than or qualitatively different from those obtained in randomized, controlled trials. (N Engl J Med 2000;342: 1878-86.)},
	language = {en},
	journal = {The New England Journal of Medicine},
	author = {Benson, Kjell},
	year = {2000},
	file = {Benson - 2000 - A Comparison of Observational Studies and Randomiz.pdf:/Users/jacquelinemaasch/Zotero/storage/FZI425MU/Benson - 2000 - A Comparison of Observational Studies and Randomiz.pdf:application/pdf},
}

@article{schnitzer_variable_2016,
	title = {Variable {Selection} for {Confounder} {Control}, {Flexible} {Modeling} and {Collaborative} {Targeted} {Minimum} {Loss}-{Based} {Estimation} in {Causal} {Inference}},
	volume = {12},
	issn = {2194-573X, 1557-4679},
	url = {https://www.degruyter.com/document/doi/10.1515/ijb-2015-0017/html},
	doi = {10.1515/ijb-2015-0017},
	abstract = {This paper investigates the appropriateness of the integration of flexible propensity score modeling (nonparametric or machine learning approaches) in semiparametric models for the estimation of a causal quantity, such as the mean outcome under treatment. We begin with an overview of some of the issues involved in knowledge-based and statistical variable selection in causal inference and the potential pitfalls of automated selection based on the fit of the propensity score. Using a simple example, we directly show the consequences of adjusting for pure causes of the exposure when using inverse probability of treatment weighting (IPTW). Such variables are likely to be selected when using a naive approach to model selection for the propensity score. We describe how the method of Collaborative Targeted minimum loss-based estimation (C-TMLE; van der Laan and Gruber, 2010) capitalizes on the collaborative double robustness property of semiparametric efficient estimators to select covariates for the propensity score based on the error in the conditional outcome model. Finally, we compare several approaches to automated variable selection in low-and high-dimensional settings through a simulation study. From this simulation study, we conclude that using IPTW with flexible prediction for the propensity score can result in inferior estimation, while Targeted minimum loss-based estimation and C-TMLE may benefit from flexible prediction and remain robust to the presence of variables that are highly correlated with treatment. However, in our study, standard influence function-based methods for the variance underestimated the standard errors, resulting in poor coverage under certain data-generating scenarios.},
	language = {en},
	number = {1},
	urldate = {2023-07-30},
	journal = {The International Journal of Biostatistics},
	author = {Schnitzer, Mireille E. and Lok, Judith J. and Gruber, Susan},
	month = may,
	year = {2016},
	pages = {97--115},
	file = {Schnitzer et al. - 2016 - Variable Selection for Confounder Control, Flexibl.pdf:/Users/jacquelinemaasch/Zotero/storage/QZWFELWF/Schnitzer et al. - 2016 - Variable Selection for Confounder Control, Flexibl.pdf:application/pdf},
}

@article{deaton_understanding_2018,
	title = {Understanding and misunderstanding randomized controlled trials},
	volume = {210},
	issn = {02779536},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0277953617307359},
	doi = {10.1016/j.socscimed.2017.12.005},
	abstract = {Randomized Controlled Trials (RCTs) are increasingly popular in the social sciences, not only in medicine. We argue that the lay public, and sometimes researchers, put too much trust in RCTs over other methods of investigation. Contrary to frequent claims in the applied literature, randomization does not equalize everything other than the treatment in the treatment and control groups, it does not automatically deliver a precise estimate of the average treatment eﬀect (ATE), and it does not relieve us of the need to think about (observed or unobserved) covariates. Finding out whether an estimate was generated by chance is more diﬃcult than commonly believed. At best, an RCT yields an unbiased estimate, but this property is of limited practical value. Even then, estimates apply only to the sample selected for the trial, often no more than a convenience sample, and justiﬁcation is required to extend the results to other groups, including any population to which the trial sample belongs, or to any individual, including an individual in the trial. Demanding ‘external validity’ is unhelpful because it expects too much of an RCT while undervaluing its potential contribution. RCTs do indeed require minimal assumptions and can operate with little prior knowledge. This is an advantage when persuading distrustful audiences, but it is a disadvantage for cumulative scientiﬁc progress, where prior knowledge should be built upon, not discarded. RCTs can play a role in building scientiﬁc knowledge and useful predictions but they can only do so as part of a cumulative program, combining with other methods, including conceptual and theoretical development, to discover not ‘what works’, but ‘why things work’.},
	language = {en},
	urldate = {2023-07-30},
	journal = {Social Science \& Medicine},
	author = {Deaton, Angus and Cartwright, Nancy},
	month = aug,
	year = {2018},
	pages = {2--21},
	file = {Deaton and Cartwright - 2018 - Understanding and misunderstanding randomized cont.pdf:/Users/jacquelinemaasch/Zotero/storage/2538J5SN/Deaton and Cartwright - 2018 - Understanding and misunderstanding randomized cont.pdf:application/pdf},
}

@article{rubin_estimating_1974,
	title = {Estimating causal effects of treatments in randomized and nonrandomized studies.},
	volume = {66},
	issn = {1939-2176, 0022-0663},
	url = {http://doi.apa.org/getdoi.cfm?doi=10.1037/h0037350},
	doi = {10.1037/h0037350},
	language = {en},
	number = {5},
	urldate = {2023-07-30},
	journal = {Journal of Educational Psychology},
	author = {Rubin, Donald B.},
	month = oct,
	year = {1974},
	pages = {688--701},
	file = {Rubin - 1974 - Estimating causal effects of treatments in randomi.pdf:/Users/jacquelinemaasch/Zotero/storage/57NVZGZH/Rubin - 1974 - Estimating causal effects of treatments in randomi.pdf:application/pdf},
}

@article{brookhart_variable_2006,
	title = {Variable {Selection} for {Propensity} {Score} {Models}},
	volume = {163},
	issn = {1476-6256, 0002-9262},
	url = {http://academic.oup.com/aje/article/163/12/1149/97130/Variable-Selection-for-Propensity-Score-Models},
	doi = {10.1093/aje/kwj149},
	language = {en},
	number = {12},
	urldate = {2023-07-30},
	journal = {American Journal of Epidemiology},
	author = {Brookhart, M. Alan and Schneeweiss, Sebastian and Rothman, Kenneth J. and Glynn, Robert J. and Avorn, Jerry and Stürmer, Til},
	month = jun,
	year = {2006},
	pages = {1149--1156},
	file = {Brookhart et al. - 2006 - Variable Selection for Propensity Score Models.pdf:/Users/jacquelinemaasch/Zotero/storage/UU35NEDV/Brookhart et al. - 2006 - Variable Selection for Propensity Score Models.pdf:application/pdf},
}

@article{haggstrom_covsel_2015,
	title = {{CovSel} : {An} {R} {Package} for {Covariate} {Selection} {When} {Estimating} {Average} {Causal} {Effects}},
	volume = {68},
	issn = {1548-7660},
	shorttitle = {\textbf{{CovSel}}},
	url = {http://www.jstatsoft.org/v68/i01/},
	doi = {10.18637/jss.v068.i01},
	language = {en},
	number = {1},
	urldate = {2023-07-30},
	journal = {Journal of Statistical Software},
	author = {Häggström, Jenny and Persson, Emma and Waernbaum, Ingeborg and De Luna, Xavier},
	year = {2015},
	file = {Häggström et al. - 2015 - CovSel  An R Package for Covariate .pdf:/Users/jacquelinemaasch/Zotero/storage/U3TRVAV8/Häggström et al. - 2015 - CovSel  An R Package for Covariate .pdf:application/pdf},
}

@article{tsamardinos_max-min_2006,
	title = {The max-min hill-climbing {Bayesian} network structure learning algorithm},
	volume = {65},
	issn = {0885-6125, 1573-0565},
	doi = {10.1007/s10994-006-6889-7},
	abstract = {We present a new algorithm for Bayesian network structure learning, called Max-Min Hill-Climbing (MMHC). The algorithm combines ideas from local learning, constraint-based, and search-and-score techniques in a principled and effective way. It ﬁrst reconstructs the skeleton of a Bayesian network and then performs a Bayesian-scoring greedy hill-climbing search to orient the edges. In our extensive empirical evaluation MMHC outperforms on average and in terms of various metrics several prototypical and state-of-the-art algorithms, namely the PC, Sparse Candidate, Three Phase Dependency Analysis, Optimal Reinsertion, Greedy Equivalence Search, and Greedy Search. These are the ﬁrst empirical results simultaneously comparing most of the major Bayesian network algorithms against each other. MMHC offers certain theoretical advantages, speciﬁcally over the Sparse Candidate algorithm, corroborated by our experiments. MMHC and detailed results of our study are publicly available at http://www.dsl-lab.org/supplements/mmhc paper/mmhc index.html.},
	language = {en},
	number = {1},
	urldate = {2023-07-30},
	journal = {Machine Learning},
	author = {Tsamardinos, Ioannis and Brown, Laura E. and Aliferis, Constantin F.},
	month = oct,
	year = {2006},
	pages = {31--78},
	file = {Tsamardinos et al. - 2006 - The max-min hill-climbing Bayesian network structu.pdf:/Users/jacquelinemaasch/Zotero/storage/RYJW62XE/Tsamardinos et al. - 2006 - The max-min hill-climbing Bayesian network structu.pdf:application/pdf},
}

@article{elwert_endogenous_2014,
	title = {Endogenous {Selection} {Bias}: {The} {Problem} of {Conditioning} on a {Collider} {Variable}},
	volume = {40},
	issn = {0360-0572, 1545-2115},
	shorttitle = {Endogenous {Selection} {Bias}},
	url = {https://www.annualreviews.org/doi/10.1146/annurev-soc-071913-043455},
	doi = {10.1146/annurev-soc-071913-043455},
	abstract = {Endogenous selection bias is a central problem for causal inference. Recognizing the problem, however, can be difﬁcult in practice. This article introduces a purely graphical way of characterizing endogenous selection bias and of understanding its consequences (Herna´n et al. 2004). We use causal graphs (direct acyclic graphs, or DAGs) to highlight that endogenous selection bias stems from conditioning (e.g., controlling, stratifying, or selecting) on a so-called collider variable, i.e., a variable that is itself caused by two other variables, one that is (or is associated with) the treatment and another that is (or is associated with) the outcome. Endogenous selection bias can result from direct conditioning on the outcome variable, a post-outcome variable, a post-treatment variable, and even a pre-treatment variable. We highlight the difference between endogenous selection bias, common-cause confounding, and overcontrol bias and discuss numerous examples from social stratiﬁcation, cultural sociology, social network analysis, political sociology, social demography, and the sociology of education.},
	language = {en},
	number = {1},
	urldate = {2023-07-30},
	journal = {Annual Review of Sociology},
	author = {Elwert, Felix and Winship, Christopher},
	month = jul,
	year = {2014},
	pages = {31--53},
	file = {Elwert and Winship - 2014 - Endogenous Selection Bias The Problem of Conditio.pdf:/Users/jacquelinemaasch/Zotero/storage/4TL8STFV/Elwert and Winship - 2014 - Endogenous Selection Bias The Problem of Conditio.pdf:application/pdf},
}

@article{cheng_local_2022,
	title = {Local search for efficient causal effect estimation},
	issn = {1041-4347, 1558-2191, 2326-3865},
	url = {https://ieeexplore.ieee.org/document/9932683/},
	doi = {10.1109/TKDE.2022.3218131},
	abstract = {Causal effect estimation from observational data is a challenging problem, especially with high dimensional data and in the presence of unobserved variables. The available data-driven methods for tackling the problem either provide an estimation of the bounds of a causal effect (i.e. nonunique estimation) or have low efficiency. The major hurdle for achieving high efficiency while trying to obtain unique and unbiased causal effect estimation is how to find a proper adjustment set for confounding control in a fast way, given the huge covariate space and considering unobserved variables. In this paper, we approach the problem as a local search task for finding valid adjustment sets in data. We establish the theorems to support the local search for adjustment sets, and we show that unique and unbiased estimation can be achieved from observational data even when there exist unobserved variables. We then propose a data-driven algorithm that is fast and consistent under mild assumptions. We also make use of a frequent pattern mining method to further speed up the search of minimal adjustment sets for causal effect estimation. Experiments conducted on extensive synthetic and real-world datasets demonstrate that the proposed algorithm outperforms the state-of-the-art criteria/estimators in both accuracy and time-efficiency.},
	language = {en},
	urldate = {2023-07-30},
	journal = {IEEE Transactions on Knowledge and Data Engineering},
	author = {Cheng, Debo and Li, Jiuyong and Liu, Lin and Zhang, Jiji and Liu, Jixue and Le, Thuc Duy},
	year = {2022},
	pages = {1--14},
	file = {Cheng et al. - 2022 - Local search for efficient causal effect estimatio.pdf:/Users/jacquelinemaasch/Zotero/storage/UIA9237Y/Cheng et al. - 2022 - Local search for efficient causal effect estimatio.pdf:application/pdf},
}

@article{tian_evaluating_2018,
	title = {Evaluating large-scale propensity score performance through real-world and synthetic data experiments},
	volume = {47},
	issn = {0300-5771, 1464-3685},
	url = {https://academic.oup.com/ije/article/47/6/2005/5043131},
	doi = {10.1093/ije/dyy120},
	abstract = {Background: Propensity score adjustment is a popular approach for confounding control in observational studies. Reliable frameworks are needed to determine relative propensity score performance in large-scale studies, and to establish optimal propensity score model selection methods.
Methods: We detail a propensity score evaluation framework that includes synthetic and real-world data experiments. Our synthetic experimental design extends the ‘plasmode’ framework and simulates survival data under known effect sizes, and our real-world experiments use a set of negative control outcomes with presumed null effect sizes. In reproductions of two published cohort studies, we compare two propensity score estimation methods that contrast in their model selection approach: L1-regularized regression that conducts a penalized likelihood regression, and the ‘high-dimensional propensity score’ (hdPS) that employs a univariate covariate screen. We evaluate methods on a range of outcome-dependent and outcome-independent metrics.
Results: L1-regularization propensity score methods achieve superior model ﬁt, covariate balance and negative control bias reduction compared with the hdPS. Simulation results are mixed and ﬂuctuate with simulation parameters, revealing a limitation of simulation under the proportional hazards framework. Including regularization with the hdPS reduces commonly reported non-convergence issues but has little effect on propensity score performance.
Conclusions: L1-regularization incorporates all covariates simultaneously into the propensity score model and offers propensity score performance superior to the hdPS marginal screen.},
	language = {en},
	number = {6},
	urldate = {2023-07-30},
	journal = {International Journal of Epidemiology},
	author = {Tian, Yuxi and Schuemie, Martijn J and Suchard, Marc A},
	month = dec,
	year = {2018},
	pages = {2005--2014},
	file = {Tian et al. - 2018 - Evaluating large-scale propensity score performanc.pdf:/Users/jacquelinemaasch/Zotero/storage/R36WPXIP/Tian et al. - 2018 - Evaluating large-scale propensity score performanc.pdf:application/pdf},
}

@article{nandy_high-dimensional_2018,
	title = {High-dimensional consistency in score-based and hybrid structure learning},
	volume = {46},
	issn = {0090-5364},
	url = {https://projecteuclid.org/journals/annals-of-statistics/volume-46/issue-6A/High-dimensional-consistency-in-score-based-and-hybrid-structure-learning/10.1214/17-AOS1654.full},
	doi = {10.1214/17-AOS1654},
	language = {en},
	number = {6A},
	urldate = {2023-08-01},
	journal = {The Annals of Statistics},
	author = {Nandy, Preetam and Hauser, Alain and Maathuis, Marloes H.},
	month = dec,
	year = {2018},
	file = {Nandy et al. - 2018 - High-dimensional consistency in score-based and hy.pdf:/Users/jacquelinemaasch/Zotero/storage/SQR924ZA/Nandy et al. - 2018 - High-dimensional consistency in score-based and hy.pdf:application/pdf},
}

@misc{assaad_hybrids_2023,
	title = {Hybrids of {Constraint}-based and {Noise}-based {Algorithms} for {Causal} {Discovery} from {Time} {Series}},
	url = {http://arxiv.org/abs/2306.08765},
	abstract = {Constraint-based and noise-based methods have been proposed to discover summary causal graphs from observational time series under strong assumptions which can be violated or impossible to verify in real applications. Recently, a hybrid method [5] that combines these two approaches, proved to be robust to assumption violation. However, this method assumes that the summary causal graph is acyclic, but cycles are common in many applications. For example, in ecological communities, there may be cyclic relationships between predator and prey populations, creating feedback loops. Therefore, this paper presents two new frameworks for hybrids of constraint-based and noise-based methods that can discover summary causal graphs that may or may not contain cycles. For each framework, we provide two hybrid algorithms which are experimentally tested on simulated data, realistic ecological data, and real data from various applications. Experiments show that our hybrid approaches are robust and yield good results over most datasets.},
	language = {en},
	urldate = {2023-08-01},
	publisher = {arXiv},
	author = {Assaad, Charles K. and Bystrova, Daria and Arbel, Julyan and Devijver, Emilie and Gaussier, Eric and Thuiller, Wilfried},
	month = jun,
	year = {2023},
	note = {arXiv:2306.08765 [cs]},
	keywords = {Computer Science - Machine Learning, Computer Science - Artificial Intelligence},
	file = {Assaad et al. - 2023 - Hybrids of Constraint-based and Noise-based Algori.pdf:/Users/jacquelinemaasch/Zotero/storage/N3QMPSTX/Assaad et al. - 2023 - Hybrids of Constraint-based and Noise-based Algori.pdf:application/pdf},
}

@article{linusson_greedy_2023,
	title = {Greedy {Causal} {Discovery} {Is} {Geometric}},
	volume = {37},
	issn = {0895-4801, 1095-7146},
	url = {https://epubs.siam.org/doi/10.1137/21M1457205},
	doi = {10.1137/21M1457205},
	abstract = {Finding a directed acyclic graph (DAG) that best encodes the conditional independence statements observable from data is a central question within causality. Algorithms that greedily transform one candidate DAG into another given a fixed set of moves have been particularly successful, for example, the greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing algorithms. In 2010, Studeny{\textbackslash}', Hemmecke, and Lindner introduced the characteristic imset (CIM) polytope, CIMp, whose vertices correspond to Markov equivalence classes, as a way of transforming causal discovery into a linear optimization problem. We show that the moves of the aforementioned algorithms are included within classes of edges of CIMp and that restrictions placed on the skeleton of the candidate DAGs correspond to faces of CIMp. Thus, we observe that greedy equivalence search, greedy interventional equivalence search, and max-min hill climbing all have geometric realizations as greedy edge-walks along CIMp. Furthermore, the identified edges of CIMp strictly generalize the moves of these algorithms. Exploiting this generalization, we introduce a greedy simplex-type algorithm called greedy CIM , and a hybrid variant, skeletal greedy CIM , that outperforms current competitors among hybrid and constraint-based algorithms.},
	language = {en},
	number = {1},
	urldate = {2023-08-01},
	journal = {SIAM Journal on Discrete Mathematics},
	author = {Linusson, Svante and Restadh, Petter and Solus, Liam},
	month = mar,
	year = {2023},
	pages = {233--252},
	file = {Linusson et al. - 2023 - Greedy Causal Discovery Is Geometric.pdf:/Users/jacquelinemaasch/Zotero/storage/S9NPIXD9/Linusson et al. - 2023 - Greedy Causal Discovery Is Geometric.pdf:application/pdf},
}

@article{greenland_adjustments_2011,
	title = {Adjustments and their {Consequences}-{Collapsibility} {Analysis} using {Graphical} {Models}: {Adjustments} and their {Consequences}},
	volume = {79},
	issn = {03067734},
	shorttitle = {Adjustments and their {Consequences}-{Collapsibility} {Analysis} using {Graphical} {Models}},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/j.1751-5823.2011.00158.x},
	doi = {10.1111/j.1751-5823.2011.00158.x},
	abstract = {We review probabilistic and graphical rules for detecting situations in which a dependence of one variable on another is altered by adjusting for a third variable (i.e., non-collapsibility or noninvariance under adjustment), whether that dependence is causal or purely predictive. We focus on distinguishing situations in which adjustment will reduce, increase, or leave unchanged the degree of bias in an association that is taken to represent a causal effect of one variable on the other. We then consider situations in which adjustment may partially remove or introduce a bias in estimating causal effects, and some additional special cases useful for case-control studies, cohort studies with loss, and trials with non-compliance (non-adherence).},
	language = {en},
	number = {3},
	urldate = {2023-08-01},
	journal = {International Statistical Review},
	author = {Greenland, Sander and Pearl, Judea},
	month = dec,
	year = {2011},
	pages = {401--426},
	file = {Greenland and Pearl - 2011 - Adjustments and their Consequences-Collapsibility .pdf:/Users/jacquelinemaasch/Zotero/storage/6QZKZTXM/Greenland and Pearl - 2011 - Adjustments and their Consequences-Collapsibility .pdf:application/pdf},
}

@article{pethel_exact_2014,
	title = {Exact {Test} of {Independence} {Using} {Mutual} {Information}},
	volume = {16},
	issn = {1099-4300},
	url = {http://www.mdpi.com/1099-4300/16/5/2839},
	doi = {10.3390/e16052839},
	abstract = {Using a recently discovered method for producing random symbol sequences with prescribed transition counts, we present an exact null hypothesis signiﬁcance test (NHST) for mutual information between two random variables, the null hypothesis being that the mutual information is zero (i.e., independence). The exact tests reported in the literature assume that data samples for each variable are sequentially independent and identically distributed (iid). In general, time series data have dependencies (Markov structure) that violate this condition. The algorithm given in this paper is the ﬁrst exact signiﬁcance test of mutual information that takes into account the Markov structure. When the Markov order is not known or indeﬁnite, an exact test is used to determine an effective Markov order.},
	language = {en},
	number = {5},
	urldate = {2023-08-01},
	journal = {Entropy},
	author = {Pethel, Shawn and Hahs, Daniel},
	month = may,
	year = {2014},
	pages = {2839--2849},
	file = {Pethel and Hahs - 2014 - Exact Test of Independence Using Mutual Informatio.pdf:/Users/jacquelinemaasch/Zotero/storage/F57H29VT/Pethel and Hahs - 2014 - Exact Test of Independence Using Mutual Informatio.pdf:application/pdf},
}

@inproceedings{chickering_selective_2015,
	address = {Amsterdam, Netherlands},
	title = {Selective {Greedy} {Equivalence} {Search}: {Finding} {Optimal} {Bayesian} {Networks} {Using} a {Polynomial} {Number} of {Score} {Evaluations}},
	abstract = {We introduce Selective Greedy Equivalence Search (SGES), a restricted version of Greedy Equivalence Search (GES). SGES retains the asymptotic correctness of GES but, unlike GES, has polynomial performance guarantees. In particular, we show that when data are sampled independently from a distribution that is perfect with respect to a DAG G deﬁned over the observable variables then, in the limit of large data, SGES will identify the equivalence class of G after a number of score evaluations that is (1) polynomial in the number of nodes and (2) exponential in various complexity measures including maximum-number-of-parents, maximum-cliquesize, and a new measure called v-width that is at least as small as—and potentially much smaller than—the other two. More generally, we show that for any hereditary and equivalenceinvariant property Π known to hold in G, we retain the large-sample optimality guarantees of GES even if we ignore any GES deletion operator during the backward phase that results in a state for which Π does not hold in the commondescendants subgraph.},
	language = {en},
	booktitle = {Proceedings of the {Thirty} {First} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Chickering, David Maxwell and Meek, Christopher},
	year = {2015},
	file = {Chickering and Meek - Selective Greedy Equivalence Search Finding Optim.pdf:/Users/jacquelinemaasch/Zotero/storage/FN7VIRXT/Chickering and Meek - Selective Greedy Equivalence Search Finding Optim.pdf:application/pdf},
}

@misc{deb_measuring_2020,
	title = {Measuring {Association} on {Topological} {Spaces} {Using} {Kernels} and {Geometric} {Graphs}},
	url = {http://arxiv.org/abs/2010.01768},
	abstract = {In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables \$X\$ and \$Y\$ taking values in general topological spaces. These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces -- capture the strength of dependence between \$X\$ and \$Y\$ and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of graph functionals which include \$k\$-nearest neighbor graphs and minimum spanning trees. Moreover, a sub-class of these estimators are also shown to adapt to the intrinsic dimensionality of the underlying distribution. Some of these empirical measures can also be computed in near linear time. Under the hypothesis of independence between \$X\$ and \$Y\$, these empirical measures (properly normalized) have a standard normal limiting distribution. Thus, these measures can also be readily used to test the hypothesis of mutual independence between \$X\$ and \$Y\$. In fact, as far as we are aware, these are the only procedures that possess all the above mentioned desirable properties. Furthermore, when restricting to Euclidean spaces, we can make these sample measures of association finite-sample distribution-free, under the hypothesis of independence, by using multivariate ranks defined via the theory of optimal transport. The recent correlation coefficient proposed in Dette et al. (2013), Chatterjee (2019), and Azadkia and Chatterjee (2019) can be seen as a special case of this general class of measures.},
	language = {en},
	urldate = {2023-08-02},
	publisher = {arXiv},
	author = {Deb, Nabarun and Ghosal, Promit and Sen, Bodhisattva},
	month = oct,
	year = {2020},
	note = {arXiv:2010.01768 [math, stat]},
	keywords = {Mathematics - Statistics Theory, 62G10, 62H20, 60F05, 60D05, Mathematics - Probability},
	file = {Deb et al. - 2020 - Measuring Association on Topological Spaces Using .pdf:/Users/jacquelinemaasch/Zotero/storage/D7FX4SFB/Deb et al. - 2020 - Measuring Association on Topological Spaces Using .pdf:application/pdf},
}

@incollection{oliver_mixed_2021,
	address = {Cham},
	title = {A {Mixed} {Noise} and {Constraint}-{Based} {Approach} to {Causal} {Inference} in {Time} {Series}},
	volume = {12975},
	isbn = {978-3-030-86485-9 978-3-030-86486-6},
	url = {https://link.springer.com/10.1007/978-3-030-86486-6_28},
	abstract = {We address, in the context of time series, the problem of learning a summary causal graph from observations through a model with independent and additive noise. The main algorithm we propose is a hybrid method that combines the well-known constraint-based framework for causal graph discovery and the noise-based framework that gained much attention in recent years. Our method is divided into two steps. First, it uses a noise-based procedure to ﬁnd the potential causes of each time series. Then, it uses a constraint-based approach to prune all unnecessary causes. A major contribution of this study is to extend the standard causation entropy measure to time series to handle lags bigger than one time step, and to rely on a lighter version of the faithfulness hypothesis, namely the adjacency faithfulness. Experiments conducted on both simulated and real-world time series show that our approach is fast and robust wrt to diﬀerent causal structures and yields good results over all datasets, whereas previously proposed approaches tend to yield good results on only few datasets.},
	language = {en},
	urldate = {2023-08-02},
	booktitle = {Machine {Learning} and {Knowledge} {Discovery} in {Databases}. {Research} {Track}},
	publisher = {Springer International Publishing},
	author = {Assaad, Charles K. and Devijver, Emilie and Gaussier, Eric and Ait-Bachir, Ali},
	editor = {Oliver, Nuria and Pérez-Cruz, Fernando and Kramer, Stefan and Read, Jesse and Lozano, Jose A.},
	year = {2021},
	doi = {10.1007/978-3-030-86486-6_28},
	note = {Series Title: Lecture Notes in Computer Science},
	pages = {453--468},
	file = {Assaad et al. - 2021 - A Mixed Noise and Constraint-Based Approach to Cau.pdf:/Users/jacquelinemaasch/Zotero/storage/BEBRFH7Z/Assaad et al. - 2021 - A Mixed Noise and Constraint-Based Approach to Cau.pdf:application/pdf},
}

@article{zanga_survey_2022,
	title = {A {Survey} on {Causal} {Discovery}: {Theory} and {Practice}},
	volume = {151},
	issn = {0888613X},
	shorttitle = {A {Survey} on {Causal} {Discovery}},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X22001402},
	doi = {10.1016/j.ijar.2022.09.004},
	language = {en},
	urldate = {2023-08-02},
	journal = {International Journal of Approximate Reasoning},
	author = {Zanga, Alessio and Ozkirimli, Elif and Stella, Fabio},
	month = dec,
	year = {2022},
	pages = {101--129},
	file = {Zanga et al. - 2022 - A Survey on Causal Discovery Theory and Practice.pdf:/Users/jacquelinemaasch/Zotero/storage/4G9JGHB3/Zanga et al. - 2022 - A Survey on Causal Discovery Theory and Practice.pdf:application/pdf},
}

@article{ahuja_interventional_nodate,
	title = {Interventional {Causal} {Representation} {Learning}},
	abstract = {Causal representation learning seeks to extract high-level latent factors from low-level sensory data. Most existing methods rely on observational data and structural assumptions (e.g., conditional independence) to identify the latent factors. However, interventional data is prevalent across applications. Can interventional data facilitate causal representation learning? We explore this question in this paper. The key observation is that interventional data often carries geometric signatures of the latent factors’ support (i.e. what values each latent can possibly take). For example, when the latent factors are causally connected, interventions can break the dependency between the intervened latents’ support and their ancestors’. Leveraging this fact, we prove that the latent causal factors can be identified up to permutation and scaling given data from perfect do interventions. Moreover, we can achieve block affine identification, namely the estimated latent factors are only entangled with a few other latents if we have access to data from imperfect interventions. These results highlight the unique power of interventional data in causal representation learning; they can enable provable identification of latent factors without any assumptions about their distributions or dependency structure.},
	language = {en},
	author = {Ahuja, Kartik and Mahajan, Divyat and Wang, Yixin and Bengio, Yoshua},
	file = {Ahuja et al. - Interventional Causal Representation Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/ZZCDR8K7/Ahuja et al. - Interventional Causal Representation Learning.pdf:application/pdf},
}

@article{dominguez-olmedo_data_nodate,
	title = {On {Data} {Manifolds} {Entailed} by {Structural} {Causal} {Models}},
	abstract = {The geometric structure of data is an important inductive bias in machine learning. In this work, we characterize the data manifolds entailed by structural causal models. The strengths of the proposed framework are twofold: firstly, the geometric structure of the data manifolds is causally informed, and secondly, it enables causal reasoning about the data manifolds in an interventional and a counterfactual sense. We showcase the versatility of the proposed framework by applying it to the generation of causally-grounded counterfactual explanations for machine learning classifiers, measuring distances along the data manifold in a differential geometric-principled manner.},
	language = {en},
	author = {Dominguez-Olmedo, Ricardo and Karimi, Amir-Hossein and Arvanitidis, Georgios and Schölkopf, Bernhard},
	file = {Dominguez-Olmedo et al. - On Data Manifolds Entailed by Structural Causal Mo.pdf:/Users/jacquelinemaasch/Zotero/storage/CMQ8YRMF/Dominguez-Olmedo et al. - On Data Manifolds Entailed by Structural Causal Mo.pdf:application/pdf},
}

@article{quinzan_drcfs_nodate,
	title = {{DRCFS}: {Doubly} {Robust} {Causal} {Feature} {Selection}},
	abstract = {Knowing the features of a complex system that are highly relevant to a particular target variable is of fundamental interest in many areas of science. Existing approaches are often limited to linear settings, sometimes lack guarantees, and in most cases, do not scale to the problem at hand, in particular to images. We propose DRCFS, a doubly robust feature selection method for identifying the causal features even in nonlinear and high dimensional settings. We provide theoretical guarantees, illustrate necessary conditions for our assumptions, and perform extensive experiments across a wide range of simulated and semisynthetic datasets. DRCFS significantly outperforms existing state-of-the-art methods, selecting robust features even in challenging highly nonlinear and high-dimensional problems.},
	language = {en},
	author = {Quinzan, Francesco and Soleymani, Ashkan and Jaillet, Patrick and Rojas, Cristian R and Bauer, Stefan},
	file = {Quinzan et al. - DRCFS Doubly Robust Causal Feature Selection.pdf:/Users/jacquelinemaasch/Zotero/storage/86L5J7Y4/Quinzan et al. - DRCFS Doubly Robust Causal Feature Selection.pdf:application/pdf},
}

@article{deleu_joint_nodate,
	title = {Joint {Bayesian} {Inference} of {Graphical} {Structure} and {Parameterswith} a {Single} {Generative} {Flow} {Network}},
	abstract = {Generative Flow Networks (GFlowNets), a class of generative models over discrete and structured sample spaces, have been previously applied to the problem of inferring the marginal posterior distribution over the directed acyclic graph (DAG) of a Bayesian Network, given observations. Based on recent advances extending this framework to non-discrete sample spaces, we propose in this paper to approximate the joint posterior over not only the structure of a Bayesian Network, but also the parameters of its conditional probability distributions. We use a single GFlowNet whose sampling policy follows a two-phase process: the DAG is first generated sequentially one edge at a time, and then the corresponding parameters are picked once the full structure is known. Since the parameters are included in the posterior distribution, this leaves more flexibility for the local probability models of the Bayesian Network, making our approach applicable even to non-linear models parametrized by neural networks. We show that our method, called JSP-GFN, offers an accurate approximation of the joint posterior, while comparing favorably against existing methods on both simulated and real data.},
	language = {en},
	author = {Deleu, Tristan and Nishikawa-Toomey, Mizu and Subramanian, Jithendaraa and Malkin, Nikolay and Charlin, Laurent and Bengio, Yoshua},
	file = {Deleu et al. - Joint Bayesian Inference of Graphical Structure an.pdf:/Users/jacquelinemaasch/Zotero/storage/4KMFLGY3/Deleu et al. - Joint Bayesian Inference of Graphical Structure an.pdf:application/pdf},
}

@article{emezue_benchmarking_nodate,
	title = {Benchmarking {Bayesian} {Causal} {Discovery} {Methodsfor} {Downstream} {Treatment} {Effect} {Estimation}},
	abstract = {The practical utility of causality in decisionmaking is widely recognized, with causal discovery and inference being inherently intertwined. Nevertheless, a notable gap exists in the evaluation of causal discovery methods, where insufficient emphasis is placed on downstream inference. To address this gap, we evaluate six established baseline causal discovery methods and a newly proposed method based on GFlowNets, on the downstream task of treatment effect estimation. Through the implementation of a robust evaluation procedure, we offer valuable insights into the efficacy of these causal discovery methods for treatment effect estimation, considering both synthetic and real-world scenarios, as well as lowdata scenarios. Furthermore, the results of our study demonstrate that GFlowNets possess the capability to effectively capture a wide range of useful and diverse ATE modes.},
	language = {en},
	author = {Emezue, Chris Chinenye and Drouin, Alexandre and Deleu, Tristan and Bauer, Stefan and Bengio, Yoshua},
	file = {Emezue et al. - Benchmarking Bayesian Causal Discovery Methodsfor .pdf:/Users/jacquelinemaasch/Zotero/storage/AZEFL2UT/Emezue et al. - Benchmarking Bayesian Causal Discovery Methodsfor .pdf:application/pdf},
}

@article{mamaghan_diffusion_2023,
	title = {Diffusion {Based} {Causal} {Representation} {Learning}},
	abstract = {Causal reasoning can be considered a cornerstone of intelligent systems. Having access to an underlying causal graph comes with the promise of cause-effect estimation and the identification of efficient and safe interventions. However, depending on the application and the complexity of the system one causal graph might be insufficient and even the variables of interest and levels of abstractions might change. This is incompatible with currently deployed generative models including popular VAE approaches which provide only representations from a point estimate. In this work, we study recently introduced diffusion-based representations which offer access to infinite dimensional latent codes which encode different levels of information in the latent code. In a first proof of principle, we investigate the use of a single point of these infinite dimensional codes for causal representation learning and demonstrate experimentally that this approach performs comparably well in identifying the causal structure and causal variables.},
	language = {en},
	author = {Mamaghan, Amir Mohammad Karimi and Dittadi, Andrea and Bauer, Stefan and Quinzan, Francesco},
	year = {2023},
	file = {Mamaghan et al. - 2023 - Diffusion Based Causal Representation Learning.pdf:/Users/jacquelinemaasch/Zotero/storage/ZKY4LQ6F/Mamaghan et al. - 2023 - Diffusion Based Causal Representation Learning.pdf:application/pdf},
}

@misc{aglietti_constrained_2023,
	title = {Constrained {Causal} {Bayesian} {Optimization}},
	url = {http://arxiv.org/abs/2305.20011},
	abstract = {We propose constrained causal Bayesian optimization (cCBO), an approach for finding interventions in a known causal graph that optimize a target variable under some constraints. cCBO first reduces the search space by exploiting the graph structure and, if available, an observational dataset; and then solves the restricted optimization problem by modelling target and constraint quantities using Gaussian processes and by sequentially selecting interventions via a constrained expected improvement acquisition function. We propose different surrogate models that enable to integrate observational and interventional data while capturing correlation among effects with increasing levels of sophistication. We evaluate cCBO on artificial and real-world causal graphs showing successful trade off between fast convergence and percentage of feasible interventions.},
	language = {en},
	urldate = {2023-08-03},
	publisher = {arXiv},
	author = {Aglietti, Virginia and Malek, Alan and Ktena, Ira and Chiappa, Silvia},
	month = may,
	year = {2023},
	note = {arXiv:2305.20011 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Aglietti et al. - 2023 - Constrained Causal Bayesian Optimization.pdf:/Users/jacquelinemaasch/Zotero/storage/QGAY8GIV/Aglietti et al. - 2023 - Constrained Causal Bayesian Optimization.pdf:application/pdf},
}

@misc{ribeiro_high_2023,
	title = {High {Fidelity} {Image} {Counterfactuals} with {Probabilistic} {Causal} {Models}},
	url = {http://arxiv.org/abs/2306.15764},
	abstract = {We present a general causal generative modelling framework for accurate estimation of high fidelity image counterfactuals with deep structural causal models. Estimation of interventional and counterfactual queries for high-dimensional structured variables, such as images, remains a challenging task. We leverage ideas from causal mediation analysis and advances in generative modelling to design new deep causal mechanisms for structured variables in causal models. Our experiments demonstrate that our proposed mechanisms are capable of accurate abduction and estimation of direct, indirect and total effects as measured by axiomatic soundness of counterfactuals.},
	language = {en},
	urldate = {2023-08-03},
	publisher = {arXiv},
	author = {Ribeiro, Fabio De Sousa and Xia, Tian and Monteiro, Miguel and Pawlowski, Nick and Glocker, Ben},
	month = jul,
	year = {2023},
	note = {arXiv:2306.15764 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Ribeiro et al. - 2023 - High Fidelity Image Counterfactuals with Probabili.pdf:/Users/jacquelinemaasch/Zotero/storage/WZIZSWP6/Ribeiro et al. - 2023 - High Fidelity Image Counterfactuals with Probabili.pdf:application/pdf},
}

@article{singh_attention_nodate,
	title = {Attention as {Implicit} {Structural} {Inference}},
	abstract = {Attention mechanisms play a crucial role in cognitive systems by allowing them to flexibly allocate cognitive resources. Transformers, in particular, have become a dominant architecture in machine learning, with attention as their central innovation. However, the underlying intuition and formalism of attention in Transformers is based on ideas of keys and queries in database management systems. In this work, we pursue a structural inference perspective, building upon, and bringing together, previous theoretical descriptions of attention such as; Gaussian Mixture Models, alignment mechanisms and Hopfield Networks. Specifically, we demonstrate that attention can be viewed as inference over an implicitly defined set of possible adjacency structures in a graphical model, revealing the generality of such a mechanism. This perspective unifies different attentional architectures in machine learning and suggests potential modifications and generalizations of attention. We hope by providing a new lens on attention architectures our work can guide the development of new and improved attentional mechanisms.},
	language = {en},
	author = {Singh, Ryan and Buckley, Christopher L},
	file = {Singh and Buckley - Attention as Implicit Structural Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/NT85V9VG/Singh and Buckley - Attention as Implicit Structural Inference.pdf:application/pdf},
}



@article{tang_difference--differences_nodate,
	title = {Difference-in-{Differences} {Meets} {Tree}-based {Methods}: {Heterogeneous} {Treatment} {Effects} {Estimation} with {Unmeasured} {Confounding}},
	abstract = {This study considers the estimation of conditional causal effects in the presence of unmeasured confounding for a balanced panel with treatment imposed at the last time point. To address this, we combine Difference-in-differences (DiD) and tree-based methods and propose a new identification assumption that allows for the violation of the (conditional) parallel trends assumption adopted by most existing DiD methods. Under this new assumption, we prove partial identifiability of the conditional average treatment effect on the treated group (CATT). Our proposed method estimates CATT through a tree-based causal approach, guided by a novel splitting rule that avoids model misspecification and unnecessary auxiliary parameter estimation. The splitting rule measures both the error of fitting observed data and the violation of conditional parallel trends simultaneously. We also develop an ensemble of multiple trees via gradient boosting to further enhance performance. Experimental results on both synthetic and real-world datasets validate the effectiveness of our proposed method.},
	language = {en},
	author = {Tang, Caizhi and Wang, Huiyuan and Li, Xinyu and Qing, Cui and Li, Longfei and Zhou, Jun},
	file = {Tang et al. - Difference-in-Differences Meets Tree-based Methods.pdf:/Users/jacquelinemaasch/Zotero/storage/WYCAXWL5/Tang et al. - Difference-in-Differences Meets Tree-based Methods.pdf:application/pdf},
}

@article{qiao_collaborative_nodate,
	title = {Collaborative {Causal} {Inference} with {Fair} {Incentives}},
	abstract = {Collaborative causal inference (CCI) aims to improve the estimation of the causal effect of treatment variables by utilizing data aggregated from multiple self-interested parties. Since their source data are valuable proprietary assets that can be costly or tedious to obtain, every party has to be incentivized to be willing to contribute to the collaboration, such as with a guaranteed fair and sufficiently valuable reward (than performing causal inference on its own). This paper presents a reward scheme designed using the unique statistical properties that are required by causal inference to guarantee certain desirable incentive criteria (e.g., fairness, benefit) for the parties based on their contributions. To achieve this, we propose a data valuation function to value parties’ data for CCI based on the distributional closeness of its resulting treatment effect estimate to that utilizing the aggregated data from all parties. Then, we show how to value the parties’ rewards fairly based on a modified variant of the Shapley value arising from our proposed data valuation for CCI. Finally, the Shapley fair rewards to the parties are realized in the form of improved, stochastically perturbed treatment effect estimates. We empirically demonstrate the effectiveness of our reward scheme using simulated and real-world datasets.},
	language = {en},
	author = {Qiao, Rui and Xu, Xinyi and Low, Bryan Kian Hsiang},
	file = {Qiao et al. - Collaborative Causal Inference with Fair Incentive.pdf:/Users/jacquelinemaasch/Zotero/storage/HJ6AV3YD/Qiao et al. - Collaborative Causal Inference with Fair Incentive.pdf:application/pdf},
}

@article{conte_large-scale_2020,
	title = {Large-scale clique cover of real-world networks},
	volume = {270},
	issn = {08905401},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S089054011930080X},
	doi = {10.1016/j.ic.2019.104464},
	abstract = {We revisit nonlinear factor analysis from a comparatively new perspective given by advancements in causal discovery and deep learning, introducing a framework for Neuro-Causal Factor Analysis (NCFA). Our approach is fully nonparametric: It identifies factors via latent causal discovery methods and then uses a variational autoencoder (VAE) that is constrained to abide by the Markov factorization of the distribution with respect to the learned graph. We evaluate NCFA on real and synthetic data sets, finding that it performs comparably to standard VAEs on data reconstruction tasks but with the advantages of sparser architecture, lower model complexity, and causal interpretability. Unlike traditional factor analysis methods, our NCFA method allows learning and reasoning about the latent factors underlying observed data from a justifiably causal perspective, even when the relations between factors and measurements are highly nonlinear.},
	language = {en},
	urldate = {2023-08-03},
	journal = {Information and Computation},
	author = {Conte, Alessio and Grossi, Roberto and Marino, Andrea},
	month = feb,
	year = {2020},
	pages = {104464},
	file = {Conte et al. - 2020 - Large-scale clique cover of real-world networks.pdf:/Users/jacquelinemaasch/Zotero/storage/J75BXHBF/Conte et al. - 2020 - Large-scale clique cover of real-world networks.pdf:application/pdf},
}

@article{manta_gflownets_nodate,
	title = {{GFlowNets} for {Causal} {Discovery}: an {Overview}},
	abstract = {Causal relationships underpin modern science and our ability to reason. Automatically discovering useful causal relationships can greatly accelerate scientific progress and facilitate the creation of machines that can reason like we do. Traditionally, the dominant approaches to causal discovery are statistical, such as the PC algorithm. A new area of research is integrating recent advancement in machine learning with causal discovery. We focus on a series of recent work that leverages new algorithms in deep learning for causal discovery –notably, generative flow networks (GFlowNets). We discuss the unique perspectives GFlowNets bring to causal discovery.},
	language = {en},
	author = {Manta, Cristian Dragos and Hu, Edward and Bengio, Yoshua},
	file = {Manta et al. - GFlowNets for Causal Discovery an Overview.pdf:/Users/jacquelinemaasch/Zotero/storage/HTAC9LWL/Manta et al. - GFlowNets for Causal Discovery an Overview.pdf:application/pdf},
}

@article{lousdal_introduction_2018,
	title = {An introduction to instrumental variable assumptions, validation and estimation},
	volume = {15},
	issn = {1742-7622},
	url = {https://ete-online.biomedcentral.com/articles/10.1186/s12982-018-0069-7},
	doi = {10.1186/s12982-018-0069-7},
	abstract = {The instrumental variable method has been employed within economics to infer causality in the presence of unmeasured confounding. Emphasising the parallels to randomisation may increase understanding of the underlying assumptions within epidemiology. An instrument is a variable that predicts exposure, but conditional on exposure shows no independent association with the outcome. The random assignment in trials is an example of what would be expected to be an ideal instrument, but instruments can also be found in observational settings with a naturally varying phenomenon e.g. geographical variation, physical distance to facility or physician’s preference. The fourth identifying assumption has received less attention, but is essential for the generalisability of estimated effects. The instrument identifies the group of compliers in which exposure is pseudo-randomly assigned leading to exchangeability with regard to unmeasured confounders. Underlying assumptions can only partially be tested empirically and require subject-matter knowledge. Future studies employing instruments should carefully seek to validate all four assumptions, possibly drawing on parallels to randomisation.},
	language = {en},
	number = {1},
	urldate = {2023-08-05},
	journal = {Emerging Themes in Epidemiology},
	author = {Lousdal, Mette Lise},
	month = dec,
	year = {2018},
	pages = {1},
	file = {Lousdal - 2018 - An introduction to instrumental variable assumptio.pdf:/Users/jacquelinemaasch/Zotero/storage/J34XBBXI/Lousdal - 2018 - An introduction to instrumental variable assumptio.pdf:application/pdf},
}

@article{labrecque_understanding_2018,
	title = {Understanding the {Assumptions} {Underlying} {Instrumental} {Variable} {Analyses}: a {Brief} {Review} of {Falsification} {Strategies} and {Related} {Tools}},
	volume = {5},
	issn = {2196-2995},
	shorttitle = {Understanding the {Assumptions} {Underlying} {Instrumental} {Variable} {Analyses}},
	url = {http://link.springer.com/10.1007/s40471-018-0152-1},
	doi = {10.1007/s40471-018-0152-1},
	abstract = {Purpose of Review Instrumental variable (IV) methods continue to be applied to questions ranging from genetic to social epidemiology. In the epidemiologic literature, discussion of whether the assumptions underlying IVanalyses hold is often limited to only certain assumptions and even then, arguments are mostly made using subject matter knowledge. To complement subject matter knowledge, there exist a variety of falsification strategies and other tools for weighing the plausibility of the assumptions underlying IV analyses.},
	language = {en},
	number = {3},
	urldate = {2023-08-05},
	journal = {Current Epidemiology Reports},
	author = {Labrecque, Jeremy and Swanson, Sonja A.},
	month = sep,
	year = {2018},
	pages = {214--220},
	file = {Labrecque and Swanson - 2018 - Understanding the Assumptions Underlying Instrumen.pdf:/Users/jacquelinemaasch/Zotero/storage/EGNZ59DU/Labrecque and Swanson - 2018 - Understanding the Assumptions Underlying Instrumen.pdf:application/pdf},
}

@article{hernan_instruments_2006,
	title = {Instruments for {Causal} {Inference}: {An} {Epidemiologist}'s {Dream}?},
	volume = {17},
	issn = {1044-3983},
	shorttitle = {Instruments for {Causal} {Inference}},
	url = {https://journals.lww.com/00001648-200607000-00004},
	doi = {10.1097/01.ede.0000222409.00878.37},
	abstract = {The use of instrumental variable (IV) methods is attrac tive because, even in the presence of unmeasured confounding, such methods may consistently estimate the average causal effect of an exposure on an outcome. However, for this consistent estimation to be achieved, several strong conditions must hold. We review the definition of an instrumental variable, describe the conditions re quired to obtain consistent estimates of causal effects, and explore their implications in the context of a recent application of the instrumental variables approach. We also present (1) a description of the connection between 4 causal models- counterfactuals, causal directed acyclic graphs, nonparametric structural equation models, and linear structural equation models-that have been used to describe instrumental variables methods; (2) a unified presentation of IV methods for the average causal effect in the study population through structural mean models; and (3) a discussion and new extensions of instrumental variables methods based on assumptions of monotonicity.},
	language = {en},
	number = {4},
	urldate = {2023-08-05},
	journal = {Epidemiology},
	author = {Hernán, Miguel A. and Robins, James M.},
	month = jul,
	year = {2006},
	pages = {360--372},
	file = {Hernán and Robins - 2006 - Instruments for Causal Inference An Epidemiologis.pdf:/Users/jacquelinemaasch/Zotero/storage/LTY6EAUK/Hernán and Robins - 2006 - Instruments for Causal Inference An Epidemiologis.pdf:application/pdf},
}

@article{imbens_instrumental_2014,
	title = {Instrumental variables: an econometrician's perspective},
	number = {No. w19983},
	journal = {National Bureau of Economic Research},
	author = {Imbens, Guido},
	year = {2014},
	file = {w19983.pdf:/Users/jacquelinemaasch/Zotero/storage/Y4JU7GYY/w19983.pdf:application/pdf},
}

@inproceedings{gao_local_2015,
	title = {Local {Causal} {Discovery} of {Direct} {Causes} and {Effects}},
	volume = {28},
	url = {https://proceedings.neurips.cc/paper_files/paper/2015/file/fcdf25d6e191893e705819b177cddea0-Paper.pdf},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Gao, Tian and Ji, Qiang},
	editor = {Cortes, C. and Lawrence, N. and Lee, D. and Sugiyama, M. and Garnett, R.},
	year = {2015},
	file = {Gao and Ji - Local Causal Discovery of Direct Causes and Effect.pdf:/Users/jacquelinemaasch/Zotero/storage/6I4X25SB/Gao and Ji - Local Causal Discovery of Direct Causes and Effect.pdf:application/pdf},
}

@article{hoyer_causal_nodate,
	title = {Causal discovery of linear acyclic models with arbitrary distributions},
	abstract = {An important task in data analysis is the discovery of causal relationships between observed variables. For continuous-valued data, linear acyclic causal models are commonly used to model the data-generating process, and the inference of such models is a wellstudied problem. However, existing methods have signiﬁcant limitations. Methods based on conditional independencies (Spirtes et al. 1993; Pearl 2000) cannot distinguish between independence-equivalent models, whereas approaches purely based on Independent Component Analysis (Shimizu et al. 2006) are inapplicable to data which is partially Gaussian. In this paper, we generalize and combine the two approaches, to yield a method able to learn the model structure in many cases for which the previous methods provide answers that are either incorrect or are not as informative as possible. We give exact graphical conditions for when two distinct models represent the same family of distributions, and empirically demonstrate the power of our method through thorough simulations.},
	language = {en},
	author = {Hoyer, Patrik O and Hyvarinen, Aapo and Scheines, Richard and Spirtes, Peter and Ramsey, Joseph and Lacerda, Gustavo and Shimizu, Shohei},
	file = {Hoyer et al. - Causal discovery of linear acyclic models with arb.pdf:/Users/jacquelinemaasch/Zotero/storage/MYVV2C5Y/Hoyer et al. - Causal discovery of linear acyclic models with arb.pdf:application/pdf},
}

@article{runge_quantifying_2014,
	title = {Quantifying the {Strength} and {Delay} of {Climatic} {Interactions}: {The} {Ambiguities} of {Cross} {Correlation} and a {Novel} {Measure} {Based} on {Graphical} {Models}},
	volume = {27},
	issn = {0894-8755, 1520-0442},
	shorttitle = {Quantifying the {Strength} and {Delay} of {Climatic} {Interactions}},
	doi = {10.1175/JCLI-D-13-00159.1},
	abstract = {Lagged cross-correlation and regression analysis are commonly used to gain insights into interaction mechanisms between climatological processes, in particular to assess time delays and to quantify the strength of a mechanism. Exempliﬁed on temperature anomalies in Europe and the tropical Paciﬁc and Atlantic, the authors study lagged correlation and regressions analytically for a simple model system. A strong dependence on the inﬂuence of serial dependencies or autocorrelation is demonstrated, which can lead to misleading conclusions about time delays and also obscures a quantiﬁcation of the interaction mechanism.},
	language = {en},
	number = {2},
	urldate = {2023-08-31},
	journal = {Journal of Climate},
	author = {Runge, Jakob and Petoukhov, Vladimir and Kurths, Jürgen},
	month = jan,
	year = {2014},
	pages = {720--739},
	file = {Runge et al. - 2014 - Quantifying the Strength and Delay of Climatic Int.pdf:/Users/jacquelinemaasch/Zotero/storage/FHAH8VBU/Runge et al. - 2014 - Quantifying the Strength and Delay of Climatic Int.pdf:application/pdf},
}

@article{lauritzen_local_1988,
	title = {Local {Computations} with {Probabilities} on {Graphical} {Structures} and {Their} {Application} to {Expert} {Systems}},
	volume = {50},
	issn = {00359246},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1988.tb01721.x},
	doi = {10.1111/j.2517-6161.1988.tb01721.x},
	abstract = {A causal network is used in a number of areas as a depiction of patterns of 'influence' among sets of variables. In expert systems it is common to perform 'inference' by means of local computations on such large but sparse networks. In general, non-probabilistic methods are used to handle uncertainty when propagating the effects of evidence, and it has appeared that exact probabilistic methods are not computationally feasible. Motivated by an application in electromyography, we counter this claim by exploiting a range of local representations for the joint probability distribution, combined with topological changes to the original network termed 'marrying' and 'filling-in'. The resulting structure allows efficient algorithms for transfer between representations, providing rapid absorption and propagation of evidence. The scheme is first illustrated on a small, fictitious but challenging example, and the underlying theory and computational aspects are then discussed.},
	language = {en},
	number = {2},
	urldate = {2023-09-05},
	journal = {Journal of the Royal Statistical Society: Series B (Methodological)},
	author = {Lauritzen, S. L. and Spiegelhalter, D. J.},
	month = jan,
	year = {1988},
	pages = {157--194},
	file = {Lauritzen and Spiegelhalter - 1988 - Local Computations with Probabilities on Graphical.pdf:/Users/jacquelinemaasch/Zotero/storage/GTRQ2R7E/Lauritzen and Spiegelhalter - 1988 - Local Computations with Probabilities on Graphical.pdf:application/pdf},
}

@inproceedings{jensen_midas_1996,
	title = {{MIDAS}: {An} influence diagram for management of mildew in winter wheat},
	booktitle = {Proceedings of the {Twelfth} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	author = {Jensen, Allan Leck and Jensen, Finn Verner},
	year = {1996},
	pages = {349--356},
	file = {Jensen and Jensen - 1996 - MIDAS An influence diagram for management of mild.pdf:/Users/jacquelinemaasch/Zotero/storage/L2G632FD/Jensen and Jensen - 1996 - MIDAS An influence diagram for management of mild.pdf:application/pdf},
}

@article{domingo-enrich_compress_nodate,
	title = {Compress {Then} {Test}: {Powerful} {Kernel} {Testing} in {Near}-linear {Time}},
	abstract = {Kernel two-sample testing provides a powerful framework for distinguishing any pair of distributions based on n sample points. However, existing kernel tests either run in n2 time or sacrifice undue power to improve runtime. To address these shortcomings, we introduce Compress Then Test (CTT), a new framework for high-powered kernel testing based on sample compression. CTT cheaply approximates an expensive test by compressing each n point sample into a small but provably high-fidelity coreset. For standard kernels and subexponential distributions, CTT inherits the statistical behavior of a quadratic-time test—recovering the same optimal detection boundary—while running in near-linear time. We couple these advances with cheaper permutation testing, justified by new power analyses; improved time-vs.-quality guarantees for low-rank approximation; and a fast aggregation procedure for identifying especially discriminating kernels. In our experiments with real and simulated data, CTT and its extensions provide 20–200x speed-ups over state-of-the-art approximate MMD tests with no loss of power.},
	language = {en},
	author = {Domingo-Enrich, Carles and Dwivedi, Raaz and Mackey, Lester},
	file = {Domingo-Enrich et al. - Compress Then Test Powerful Kernel Testing in Nea.pdf:/Users/jacquelinemaasch/Zotero/storage/NISIYQ86/Domingo-Enrich et al. - Compress Then Test Powerful Kernel Testing in Nea.pdf:application/pdf},
}

@article{gretton_kernel_2007,
  title={A kernel statistical test of independence},
  author={Gretton, Arthur and Fukumizu, Kenji and Teo, Choon and Song, Le and Sch{\"o}lkopf, Bernhard and Smola, Alex},
  journal={Advances in neural information processing systems},
  volume={20},
  year={2007}
}

@article{li_constraint-based_nodate,
	title = {Constraint-based {Causal} {Structure} {Learning} with {Consistent} {Separating} {Sets}},
	abstract = {We consider constraint-based methods for causal structure learning, such as the PC algorithm or any PC-derived algorithms whose ﬁrst step consists in pruning a complete graph to obtain an undirected graph skeleton, which is subsequently oriented. All constraint-based methods perform this ﬁrst step of removing dispensable edges, iteratively, whenever a separating set and corresponding conditional independence can be found. Yet, constraint-based methods lack robustness over sampling noise and are prone to uncover spurious conditional independences in ﬁnite datasets. In particular, there is no guarantee that the separating sets identiﬁed during the iterative pruning step remain consistent with the ﬁnal graph. In this paper, we propose a simple modiﬁcation of PC and PC-derived algorithms so as to ensure that all separating sets identiﬁed to remove dispensable edges are consistent with the ﬁnal graph, thus enhancing the explainability of constraint-based methods. It is achieved by repeating the constraint-based causal structure learning scheme, iteratively, while searching for separating sets that are consistent with the graph obtained at the previous iteration. Ensuring the consistency of separating sets can be done at a limited complexity cost, through the use of block-cut tree decomposition of graph skeletons, and is found to increase their validity in terms of actual d-separation. It also signiﬁcantly improves the sensitivity of constraint-based methods while retaining good overall structure learning performance. Finally and foremost, ensuring sepset consistency improves the interpretability of constraint-based models for real-life applications.},
	language = {en},
	author = {Li, Honghao and Cabeli, Vincent and Sella, Nadir and Isambert, Hervé},
	file = {Li et al. - Constraint-based Causal Structure Learning with Co.pdf:/Users/jacquelinemaasch/Zotero/storage/UMX6BI9A/Li et al. - Constraint-based Causal Structure Learning with Co.pdf:application/pdf},
}

@misc{scutari_learning_2010,
	title = {Learning {Bayesian} {Networks} with the bnlearn {R} {Package}},
	url = {http://arxiv.org/abs/0908.3817},
	abstract = {bnlearn is an R package which includes several algorithms for learning the structure of Bayesian networks with either discrete or continuous variables. Both constraint-based and score-based algorithms are implemented, and can use the functionality provided by the snow package to improve their performance via parallel computing. Several network scores and conditional independence algorithms are available for both the learning algorithms and independent use. Advanced plotting options are provided by the Rgraphviz package.},
	language = {en},
	urldate = {2023-09-12},
	publisher = {arXiv},
	author = {Scutari, Marco},
	month = jul,
	year = {2010},
	note = {arXiv:0908.3817 [stat]},
	keywords = {Statistics - Machine Learning},
	file = {Scutari - 2010 - Learning Bayesian Networks with the bnlearn R Pack.pdf:/Users/jacquelinemaasch/Zotero/storage/8ZHTEPW6/Scutari - 2010 - Learning Bayesian Networks with the bnlearn R Pack.pdf:application/pdf},
}

@inproceedings{spirtes_anytime_2001,
	title = {An {Anytime} {Algorithm} for {Causal} {Inference}},
	volume = {R3},
	abstract = {The Fast Casual Inference (FCI) algorithm searches for features common to observationally equivalent sets of causal directed acyclic graphs. It is correct in the large sample limit with probability one even if there is a possibility of hidden variables and selection bias. In the worst case, the number of conditional independence tests performed by the algorithm grows exponentially with the number of variables in the data set. This affects both the speed of the algorithm and the accuracy of the algorithm on small samples, because tests of independence conditional on large numbers of variables have very low power. In this paper, I prove that the FCI algorithm can be interrupted at any stage and asked for output. The output from the interrupted algorithm is still correct with probability one in the large sample limit, although possibly less informative (in the sense that it answers “Can’t tell” for a larger number of questions) than if the FCI algorithm had been allowed to continue uninterrupted.},
	language = {en},
	booktitle = {Proceedings of the {Eighth} {International} {Workshop} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Spirtes, Peter},
	year = {2001},
	pages = {278--285},
	file = {Spirtes - An Anytime Algorithm for Causal Inference.pdf:/Users/jacquelinemaasch/Zotero/storage/ZZ5TT8BV/Spirtes - An Anytime Algorithm for Causal Inference.pdf:application/pdf},
}

@article{wang_discovering_2014,
	title = {Discovering and orienting the edges connected to a target variable in a {DAG} via a sequential local learning approach},
	volume = {77},
	issn = {01679473},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0167947314000802},
	doi = {10.1016/j.csda.2014.03.003},
	abstract = {Given a target variable and observational data, we propose a sequential learning approach for discovering direct cause and effect variables of the target under the causal network framework. In the approach, we start from the target, sequentially find Markov blankets of variables and learn local structures over Markov blankets until we determine the causes and effects of the target variable. Without constructing a whole network over all variables, we find only a local structure around the target. The main advantage of the proposed sequential approach is that it can greatly reduce CPU time compared with whole network learning approaches for finding the causes and effects of a given target node in a large network. The proposed approach can be applied to predict the effects of external interventions. Theoretically we show the correctness of the approach under the assumptions of faithfulness, causal sufficiency and that independencies are correctly checked. These theoretical results can also be used for learning directed acyclic graphs with latent variables.},
	language = {en},
	urldate = {2023-10-08},
	journal = {Computational Statistics \& Data Analysis},
	author = {Wang, Changzhang and Zhou, You and Zhao, Qiang and Geng, Zhi},
	month = sep,
	year = {2014},
	pages = {252--266},
	file = {Wang et al. - 2014 - Discovering and orienting the edges connected to a.pdf:/Users/jacquelinemaasch/Zotero/storage/CE2UNZAQ/Wang et al. - 2014 - Discovering and orienting the edges connected to a.pdf:application/pdf},
}

@article{tsamardinos_algorithms_nodate,
	title = {Algorithms for {Large} {Scale} {Markov} {Blanket} {Discovery}},
	abstract = {This paper presents a number of new algorithms for discovering the Markov Blanket of a target variable T from training data. The Markov Blanket can be used for variable selection for classification, for causal discovery, and for Bayesian Network learning. We introduce a low-order polynomial algorithm and several variants that soundly induce the Markov Blanket under certain broad conditions in datasets with thousands of variables and compare them to other state-of-the-art local and global methods with excellent results.},
	language = {en},
	author = {Tsamardinos, Ioannis and Aliferis, Constantin F and Statnikov, Alexander},
	file = {Tsamardinos et al. - Algorithms for Large Scale Markov Blanket Discover.pdf:/Users/jacquelinemaasch/Zotero/storage/KBZZHITJ/Tsamardinos et al. - Algorithms for Large Scale Markov Blanket Discover.pdf:application/pdf},
}

@article{soleymani_causal_2022,
	title = {Causal {Feature} {Selection} via {Orthogonal} {Search}},
	url = {http://arxiv.org/abs/2007.02938},
	abstract = {The problem of inferring the direct causal parents of a response variable among a large set of explanatory variables is of high practical importance in many disciplines. However, established approaches often scale at least exponentially with the number of explanatory variables, are difﬁcult to extend to nonlinear relationships and are difﬁcult to extend to cyclic data. Inspired by Debiased machine learning methods, we study a one-vs.-the-rest feature selection approach to discover the direct causal parent of the response. We propose an algorithm that works for purely observational data while also offering theoretical guarantees, including the case of partially nonlinear relationships possibly under the presence of cycles. As it requires only one estimation for each variable, our approach is applicable even to large graphs. We demonstrate signiﬁcant improvements compared to established approaches.},
	language = {en},
	urldate = {2023-10-08},
	journal = {Transactions on Machine Learning Research},
	author = {Soleymani, Ashkan and Raj, Anant and Bauer, Stefan and Schölkopf, Bernhard and Besserve, Michel},
	month = sep,
	year = {2022},
	note = {arXiv:2007.02938 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Mathematics - Statistics Theory},
	file = {Soleymani et al. - 2022 - Causal Feature Selection via Orthogonal Search.pdf:/Users/jacquelinemaasch/Zotero/storage/JEENC5WB/Soleymani et al. - 2022 - Causal Feature Selection via Orthogonal Search.pdf:application/pdf},
}

@article{kalisch_estimating_2007,
	title = {Estimating {High}-{Dimensional} {Directed} {Acyclic} {Graphs} with the {PC}-{Algorithm}},
	abstract = {We consider the PC-algorithm (Spirtes et al., 2000) for estimating the skeleton and equivalence class of a very high-dimensional directed acyclic graph (DAG) with corresponding Gaussian distribution. The PC-algorithm is computationally feasible and often very fast for sparse problems with many nodes (variables), and it has the attractive property to automatically achieve high computational efﬁciency as a function of sparseness of the true underlying DAG. We prove uniform consistency of the algorithm for very high-dimensional, sparse DAGs where the number of nodes is allowed to quickly grow with sample size n, as fast as O(na) for any 0 {\textless} a {\textless} ∞. The sparseness assumption is rather minimal requiring only that the neighborhoods in the DAG are of lower order than sample size n. We also demonstrate the PC-algorithm for simulated data.},
	language = {en},
	number = {8},
	journal = {Journal of Machine Learning Research},
	author = {Kalisch, Markus and Buhlmann, Peter},
	year = {2007},
	pages = {613--636},
	file = {Kalisch and Buhlmann - 2007 - Estimating High-Dimensional Directed Acyclic Graph.pdf:/Users/jacquelinemaasch/Zotero/storage/R457UMVF/Kalisch and Buhlmann - 2007 - Estimating High-Dimensional Directed Acyclic Graph.pdf:application/pdf},
}

@article{mohri_structural_2005,
	title = {Structural {Zeros} versus {Sampling} {Zeros}},
	url = {https://lanzaroark.org/docs/zero.pdf},
	abstract = {Probabilistic sequence models estimated from large corpora typically require smoothing techniques to reserve some probability mass for unobserved events. These techniques fail to distinguish between events unobserved due to sampling limitations, sampling zeros, and those unobserved due to structural reasons such as syntactic constraints, structural zeros. We investigate the use of statistical tests to determine structural zeros, to avoid assigning them probability mass and thereby improve model accuracy. Experimental results on a context-free parsing task demonstrate the usefulness of these techniques.},
	language = {en},
	author = {Mohri, Mehryar and Roark, Brian},
	year = {2005},
	file = {Mohri and Roark - Structural Zeros versus Sampling Zeros.pdf:/Users/jacquelinemaasch/Zotero/storage/GXRUFIMB/Mohri and Roark - Structural Zeros versus Sampling Zeros.pdf:application/pdf},
}

@misc{hahn_feature_2022,
	title = {Feature selection in stratification estimators of causal effects: lessons from potential outcomes, causal diagrams, and structural equations},
	shorttitle = {Feature selection in stratification estimators of causal effects},
	url = {http://arxiv.org/abs/2209.11400},
	abstract = {What is the ideal regression (if any) for estimating average causal effects? We study this question in the setting of discrete covariates, deriving expressions for the ﬁnite-sample variance of various stratiﬁcation estimators. This approach clariﬁes the fundamental statistical phenomena underlying many widely-cited results. Our exposition combines insights from three distinct methodological traditions for studying causal effect estimation: potential outcomes, causal diagrams, and structural models with additive errors. Key words and phrases: Causal inference, Dimension reduction, Propensity score, Regularized regression, Semi-supervised learning, Variable selection.},
	language = {en},
	urldate = {2023-10-16},
	publisher = {arXiv},
	author = {Hahn, P. Richard and Herren, Andrew},
	month = sep,
	year = {2022},
	note = {arXiv:2209.11400 [stat]},
	keywords = {Statistics - Machine Learning, Statistics - Methodology},
	file = {Hahn and Herren - 2022 - Feature selection in stratification estimators of .pdf:/Users/jacquelinemaasch/Zotero/storage/4MBWQ8PM/Hahn and Herren - 2022 - Feature selection in stratification estimators of .pdf:application/pdf},
}


%%% TOPO SORTS
@article{causal_ground,
    author = {Faller, Philipp and Vankadara, Leena and Mastakouri, Atalanti and Locatello, Francesco and Janzing, Dominik},
    title = {Self-Compatibility: Evaluating Causal Discovery without Ground Truth},
    journal = {International Conference on Artifical Intellgience and Statistics },
    year = {2024}
}

@inproceedings{montagna_scalable_2023,
	title = {Scalable {Causal} {Discovery} with {Score} {Matching}},
	language = {en},
	urldate = {2024-01-15},
	booktitle = {Proceedings of the 2nd {Conference} on {Causal} {Learning} and {Reasoning}},
	publisher = {arXiv},
	author = {Montagna, Francesco and Noceti, Nicoletta and Rosasco, Lorenzo and Zhang, Kun and Locatello, Francesco},
	month = apr,
	year = {2023},
	note = {arXiv:2304.03382 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	file = {Montagna et al. - 2023 - Scalable Causal Discovery with Score Matching.pdf:/Users/jacquelinemaasch/Zotero/storage/BIE98THE/Montagna et al. - 2023 - Scalable Causal Discovery with Score Matching.pdf:application/pdf},
}

@article{sanchez_diffusion_2023,
  title={Diffusion models for causal discovery via topological ordering},
  author={Sanchez, Pedro and Liu, Xiao and O'Neil, Alison Q and Tsaftaris, Sotirios A},
  journal={The Eleventh International Conference on Learning Representations (ICLR 2023)},
url = {https://arxiv.org/abs/2210.06201},
  year={2023}
}

@article{sachs2005causal,
  title={Causal protein-signaling networks derived from multiparameter single-cell data},
  author={Sachs, Karen and Perez, Omar and Pe'er, Dana and Lauffenburger, Douglas A and Nolan, Garry P},
  journal={Science},
  volume={308},
  number={5721},
  pages={523--529},
  year={2005},
  publisher={American Association for the Advancement of Science}
}

@inproceedings{montagna_causal_2023,
	title = {Causal {Discovery} with {Score} {Matching} on {Additive} {Models} with {Arbitrary} {Noise}},
	language = {en},
	urldate = {2024-01-15},
	booktitle = {Proceedings of the 2nd {Conference} on {Causal} {Learning} and {Reasoning}},
	publisher = {arXiv},
	author = {Montagna, Francesco and Noceti, Nicoletta and Rosasco, Lorenzo and Zhang, Kun and Locatello, Francesco},
	month = apr,
	year = {2023},
	note = {arXiv:2304.03265 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Montagna et al. - 2023 - Causal Discovery with Score Matching on Additive M.pdf:/Users/jacquelinemaasch/Zotero/storage/438NJU76/Montagna et al. - 2023 - Causal Discovery with Score Matching on Additive M.pdf:application/pdf},
}

@inproceedings{montagna_assumption_2023,
	title = {Assumption violations in causal discovery and the robustness of score matching},
	language = {en},
	urldate = {2024-01-15},
	booktitle = {37th {Conference} on {Neural} {Information} {Processing} {Systems}},
	author = {Montagna, Francesco and Mastakouri, Atalanti A. and Eulig, Elias and Noceti, Nicoletta and Rosasco, Lorenzo and Janzing, Dominik and Aragam, Bryon and Locatello, Francesco},
	month = oct,
	year = {2023},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Montagna et al. - 2023 - Assumption violations in causal discovery and the .pdf:/Users/jacquelinemaasch/Zotero/storage/PHMWHPTX/Montagna et al. - 2023 - Assumption violations in causal discovery and the .pdf:application/pdf},
}

@article{reisach_beware_2021,
  title={Beware of the simulated dag! causal discovery benchmarks may be easy to game},
  author={Reisach, Alexander and Seiler, Christof and Weichwald, Sebastian},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={27772--27784},
  year={2021}
}

@inproceedings{reisach_scale-invariant_2023,
	title = {A {Scale}-{Invariant} {Sorting} {Criterion} to {Find} a {Causal} {Order} in {Additive} {Noise} {Models}},
	language = {en},
	urldate = {2024-01-15},
	booktitle = {37th {Conference} on {Neural} {Information} {Processing} {Systems}},
	publisher = {arXiv},
	author = {Reisach, Alexander G. and Tami, Myriam and Seiler, Christof and Chambaz, Antoine and Weichwald, Sebastian},
	month = oct,
	year = {2023},
	note = {arXiv:2303.18211 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology},
	file = {Reisach et al. - 2023 - A Scale-Invariant Sorting Criterion to Find a Caus.pdf:/Users/jacquelinemaWe show the correctness of Algorithm \ref{algo:lin} in \ref{theorem:lin_alg} (proof in Appendix \ref{appendix:LHTS}).
\begin{theorem}\label{theorem:lin_alg}
Given a graph $G$, Algorithm \ref{algo:lin} asymptotically finds a correct hierarchical sort of $G$.
\end{theorem}asch/Zotero/storage/RIA9IM8U/Reisach et al. - 2023 - A Scale-Invariant Sorting Criterion to Find a Caus.pdf:application/pdf},
}



@article{buhlmann_cam_2014,
	title = {{CAM}: {Causal} additive models, high-dimensional order search and penalized regression},
	volume = {42},
	issn = {0090-5364},
	shorttitle = {{CAM}},
	doi = {10.1214/14-AOS1260},
	language = {en},
	number = {6},
	urldate = {2024-01-15},
	journal = {The Annals of Statistics},
	author = {Bühlmann, Peter and Peters, Jonas and Ernest, Jan},
	month = dec,
	year = {2014},
	note = {arXiv:1310.1533 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology},
	file = {Bühlmann et al. - 2014 - CAM Causal additive models, high-dimensional orde.pdf:/Users/jacquelinemaasch/Zotero/storage/5R4ETNCI/Bühlmann et al. - 2014 - CAM Causal additive models, high-dimensional orde.pdf:application/pdf},
}

@inproceedings{rolland_score_2022,
	title = {Score {Matching} {Enables} {Causal} {Discovery} of {Nonlinear} {Additive} {Noise} {Models}},
	language = {en},
	booktitle = {Proceedings of the 39 th {International} {Conference} on {Machine} {Learning}},
	author = {Rolland, Paul and Cevher, Volkan and Kleindessner, Matthaus and Russel, Chris and Scholkopf, Bernhard and Janzing, Dominik and Locatello, Francesco},
	year = {2022},
	file = {Rolland et al. - Score Matching Enables Causal Discovery of Nonline.pdf:/Users/jacquelinemaasch/Zotero/storage/YJMMZLVY/Rolland et al. - Score Matching Enables Causal Discovery of Nonline.pdf:application/pdf},
}

@misc{montagna_shortcuts_2023,
	title = {Shortcuts for causal discovery of nonlinear models by score matching},
	url = {http://arxiv.org/abs/2310.14246},
	language = {en},
	urldate = {2024-01-15},
	publisher = {arXiv},
	author = {Montagna, Francesco and Noceti, Nicoletta and Rosasco, Lorenzo and Locatello, Francesco},
	month = oct,
	year = {2023},
	note = {arXiv:2310.14246 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Methodology},
	file = {Montagna et al. - 2023 - Shortcuts for causal discovery of nonlinear models.pdf:/Users/jacquelinemaasch/Zotero/storage/RG2AL3WG/Montagna et al. - 2023 - Shortcuts for causal discovery of nonlinear models.pdf:application/pdf},
}

@misc{comon_ica_1994,
	title = {Independent component analysis - a new concept?},
	url = {https://hal.science/hal-00417283/document},
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@misc{additive_hypothesis_marra, 
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@misc{lasso_tibshirani, 
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@misc{gsp_solus_2021, 
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@misc{grasp_liam_2021, 
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@misc{discovery_review_2024, 
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@misc{sample_bound_2023, 
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Modeling}},
	author = {Zhu, Zhenyu and Locatello, Francesco and Cevher, Volkan},
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journal = {37th Conference on Neural Information Processing Systems (NeurIPS 2023)}
}


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