
@misc{manton_primer_2015,
	title = {A {Primer} on {Reproducing} {Kernel} {Hilbert} {Spaces}},
	url = {http://arxiv.org/abs/1408.0952},
	abstract = {Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces.},
	urldate = {2024-02-26},
	publisher = {arXiv},
	author = {Manton, Jonathan H. and Amblard, Pierre-Olivier},
	month = nov,
	year = {2015},
	note = {arXiv:1408.0952 [math]},
	keywords = {Mathematics - History and Overview},
}

@article{aitkin_estimation_1985,
	title = {Estimation and {Hypothesis} {Testing} in {Finite} {Mixture} {Models}},
	volume = {47},
	issn = {0035-9246},
	url = {https://www.jstor.org/stable/2345545},
	abstract = {Finite mixture models are a useful class of models for application to data. When sample sizes are not large and the number of underlying densities is in question, likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter, λ, and the parameter of the underlying densities, θ, are problematical. Our approach places a prior distribution on λ and estimates θ by maximizing the likelihood of the data given θ with λ integrated out. Advantages of this approach, computational issues using the EM algorithm and directions for further work are discussed. The technique is applied to two examples.},
	number = {1},
	urldate = {2024-02-26},
	journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
	author = {Aitkin, Murray and Rubin, Donald B.},
	year = {1985},
	note = {Publisher: [Royal Statistical Society, Wiley]},
	pages = {67--75},
}

@article{zhu_hypothesis_nodate,
	title = {Hypothesis {Testing} in {Mixture} {Regression} {Models} ({Mathematical} {Details})},
	language = {en},
	author = {Zhu, Hongtu and Zhang, Heping},
}

@inproceedings{law_testing_2017,
	title = {Testing and {Learning} on {Distributions} with {Symmetric} {Noise} {Invariance}},
	volume = {30},
	url = {https://proceedings.neurips.cc/paper_files/paper/2017/hash/67d16d00201083a2b118dd5128dd6f59-Abstract.html},
	abstract = {Kernel embeddings of distributions and the Maximum Mean Discrepancy (MMD), the resulting distance between distributions, are useful tools for fully nonparametric two-sample testing and learning on distributions. However, it is rarely that all possible differences between samples are of interest -- discovered differences can be due to different types of measurement noise, data collection artefacts or other irrelevant sources of variability. We propose distances between distributions which encode invariance to additive symmetric noise, aimed at testing whether the assumed true underlying processes differ. Moreover, we construct invariant features of distributions, leading to learning algorithms robust to the impairment of the input distributions with symmetric additive noise.},
	urldate = {2024-02-25},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Law, Ho Chung and Yau, Christopher and Sejdinovic, Dino},
	year = {2017},
}

@article{berger_encounters_2019,
	title = {Encounters with {Imprecise} {Probabilities}},
	language = {en},
	author = {Berger, Jim},
	year = {2019},
}

@article{liu_credal_nodate,
	title = {Credal classification of uncertain data based on belief function theory},
	language = {en},
	author = {Liu, Zhun-Ga},
}

@inproceedings{caprio_dynamic_2023,
	title = {Dynamic precise and imprecise probability kinematics},
	url = {https://proceedings.mlr.press/v215/caprio23a.html},
	abstract = {We introduce dynamic probability kinematics (DPK), a method for an agent to mechanically update subjective beliefs in the presence of partial information. We then generalize DPK to dynamic imprecise probability kinematics (DIPK), which allows the agent to express their initial beliefs via a set of probabilities in order to further take ambiguity into account. Examples are provided to illustrate how the methods work.},
	language = {en},
	urldate = {2024-02-15},
	booktitle = {Proceedings of the {Thirteenth} {International} {Symposium} on {Imprecise} {Probability}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Caprio, Michele and Gong, Ruobin},
	month = jul,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {72--83},
}

@article{sharafutdinov_application_2022,
	title = {Application of convex hull analysis for the evaluation of data heterogeneity between patient populations of different origin and implications of hospital bias in downstream machine-learning-based data processing: {A} comparison of 4 critical-care patient datasets},
	volume = {5},
	issn = {2624-909X},
	shorttitle = {Application of convex hull analysis for the evaluation of data heterogeneity between patient populations of different origin and implications of hospital bias in downstream machine-learning-based data processing},
	url = {https://www.frontiersin.org/articles/10.3389/fdata.2022.603429/full},
	doi = {10.3389/fdata.2022.603429},
	abstract = {Machine learning (ML) models are developed on a learning dataset covering only a small part of the data of interest. If model predictions are accurate for the learning dataset but fail for unseen data then generalization error is considered high. This problem manifests itself within all major sub-fields of ML but is especially relevant in medical applications. Clinical data structures, patient cohorts, and clinical protocols may be highly biased among hospitals such that sampling of representative learning datasets to learn ML models remains a challenge. As ML models exhibit poor predictive performance over data ranges sparsely or not covered by the learning dataset, in this study, we propose a novel method to assess their generalization capability among different hospitals based on the convex hull (CH) overlap between multivariate datasets. To reduce dimensionality effects, we used a two-step approach. First, CH analysis was applied to find mean CH coverage between each of the two datasets, resulting in an upper bound of the prediction range. Second, 4 types of ML models were trained to classify the origin of a dataset (i.e., from which hospital) and to estimate differences in datasets with respect to underlying distributions. To demonstrate the applicability of our method, we used 4 critical-care patient datasets from different hospitals in Germany and USA. We estimated the similarity of these populations and investigated whether ML models developed on one dataset can be reliably applied to another one. We show that the strongest drop in performance was associated with the poor intersection of convex hulls in the corresponding hospitals' datasets and with a high performance of ML methods for dataset discrimination. Hence, we suggest the application of our pipeline as a first tool to assess the transferability of trained models. We emphasize that datasets from different hospitals represent heterogeneous data sources, and the transfer from one database to another should be performed with utmost care to avoid implications during real-world applications of the developed models. Further research is needed to develop methods for the adaptation of ML models to new hospitals. In addition, more work should be aimed at the creation of gold-standard datasets that are large and diverse with data from varied application sites.},
	language = {en},
	urldate = {2024-02-15},
	journal = {Frontiers in Big Data},
	author = {Sharafutdinov, Konstantin and Bhat, Jayesh S. and Fritsch, Sebastian Johannes and Nikulina, Kateryna and E. Samadi, Moein and Polzin, Richard and Mayer, Hannah and Marx, Gernot and Bickenbach, Johannes and Schuppert, Andreas},
	month = oct,
	year = {2022},
	pages = {603429},
}

@misc{xie_robust_2021,
	title = {Robust {Hypothesis} {Testing} with {Wasserstein} {Uncertainty} {Sets}},
	url = {http://arxiv.org/abs/2105.14348},
	doi = {10.48550/arXiv.2105.14348},
	abstract = {We consider a data-driven robust hypothesis test where the optimal test will minimize the worst-case performance regarding distributions that are close to the empirical distributions with respect to the Wasserstein distance. This leads to a new non-parametric hypothesis testing framework based on distributionally robust optimization, which is more robust when there are limited samples for one or both hypotheses. Such a scenario often arises from applications such as health care, online change-point detection, and anomaly detection. We study the computational and statistical properties of the proposed test by presenting a tractable convex reformulation of the original infinite-dimensional variational problem exploiting Wasserstein's properties and characterizing the radii selection for the uncertainty sets. We also demonstrate the good performance of our method on synthetic and real data.},
	urldate = {2024-02-14},
	publisher = {arXiv},
	author = {Xie, Liyan and Gao, Rui and Xie, Yao},
	month = may,
	year = {2021},
	note = {arXiv:2105.14348 [math, stat]},
	keywords = {Mathematics - Statistics Theory, Statistics - Methodology},
}

@book{vejnarova_symbolic_2021,
	address = {Cham},
	series = {Lecture {Notes} in {Computer} {Science}},
	title = {Symbolic and {Quantitative} {Approaches} to {Reasoning} with {Uncertainty}: 16th {European} {Conference}, {ECSQARU} 2021, {Prague}, {Czech} {Republic}, {September} 21–24, 2021, {Proceedings}},
	volume = {12897},
	isbn = {978-3-030-86771-3 978-3-030-86772-0},
	shorttitle = {Symbolic and {Quantitative} {Approaches} to {Reasoning} with {Uncertainty}},
	url = {https://link.springer.com/10.1007/978-3-030-86772-0},
	language = {en},
	urldate = {2024-02-13},
	publisher = {Springer International Publishing},
	editor = {Vejnarová, Jiřina and Wilson, Nic},
	year = {2021},
	doi = {10.1007/978-3-030-86772-0},
}

@misc{caprio_credal_2024,
	title = {Credal {Learning} {Theory}},
	url = {http://arxiv.org/abs/2402.00957},
	abstract = {Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learnt from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not) as well as infinite model spaces, which directly generalize classical results.},
	urldate = {2024-02-13},
	publisher = {arXiv},
	author = {Caprio, Michele and Sultana, Maryam and Elia, Eleni and Cuzzolin, Fabio},
	month = feb,
	year = {2024},
	note = {arXiv:2402.00957 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{voss_partition_nodate,
	title = {Partition testing: {A} generalization of hypothesis testing},
	abstract = {We consider a variation on partition testing as introduced by Finner and Strassburger (2002), demonstrating this variation to be a useful generalization of traditional hypothesis testing. By choosing an appropriate partition of the parameter space and testing each set in the partition with a size-α test, more speciﬁc inferences are obtained compared to those provided by traditional methods of hypothesis testing, while still controlling the probability of making any false assertions to be at most α. Generalizing the traditional two-sided test of a point null hypothesis, one choice of the partition results in rejection of all values except those in the traditional two-sided conﬁdence interval, and another choice of the partition provides better directional inference at no cost whenever the traditional point null hypothesis is rejected. Generalizing the traditional one-sided test, an appropriate choice of the partition results in rejection of all values except those ‘accepted’ in view of the corresponding one-sided conﬁdence bound. In short, the proposed partition test generalizes traditional testing so as to make inferences for the partition test exactly match those obtained by the corresponding conﬁdence estimation approach.},
	language = {en},
	author = {Voss, Daniel T},
}

@article{romano_stepwise_nodate,
	title = {Stepwise {Multiple} {Testing} as {Formalized} {Data} {Snooping}},
	abstract = {It is common in econometric applications that several hypothesis tests are carried out at the same time. The problem then becomes how to decide which hypotheses to reject, accounting for the multitude of tests. This paper suggests a stepwise multiple testing procedure which asymptotically controls the familywise error rate at a desired level. Compared to related singlestep methods, the procedure is more powerful in the sense that it often will reject more false hypotheses. In addition, we advocate the use of studentization when it is feasible. Unlike some stepwise methods, the method implicitly captures the joint dependence structure of the test statistics, which results in increased ability to detect alternative hypotheses. We prove asymptotic control of the familywise error rate under minimal assumptions. The methodology is presented in the context of comparing several strategies to a common benchmark and deciding which strategies actually beat the benchmark. However, our ideas can easily be extended and/or modiﬁed to other contexts, such as making inference for the individual regression coeﬃcients in a multiple regression framework. Some simulation studies show the improvements of our methods over previous proposals. We also provide an application to a set of real data.},
	language = {en},
	author = {Romano, Joseph P and Wolf, Michael},
}

@inproceedings{malek_sequential_2017,
	title = {Sequential {Multiple} {Hypothesis} {Testing} with {Type} {I} {Error} {Control}},
	url = {https://proceedings.mlr.press/v54/malek17a.html},
	abstract = {This work studies multiple hypothesis testing in the setting when we obtain data sequentially and may choose when to stop sampling. We summarize the notion of a sequential p-value (one that can be continually updated and still maintain a type I error guarantee) and provide several examples from the literature. This tool allows us to convert step-up or step-down multiple hypothesis testing procedures in the fixed-horizon setting (which includes Benjamini-Hochberg, Holm, and Bonferroni)  into sequential versions that allow the statistician to reject a hypothesis as soon as the sequential p-value reaches a threshold. We show that if the original procedure has a type I error guarantee in a certain family (including FDR and FWER), then the sequential conversion inherits an analogous guarantee. The conversion also allows for allocating samples in a data-dependent way, and we provide simulated experiments demonstrating an increased number of rejections when compared to the fixed-horizon setting.},
	language = {en},
	urldate = {2024-02-12},
	booktitle = {Proceedings of the 20th {International} {Conference} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Malek, Alan and Katariya, Sumeet and Chow, Yinlam and Ghavamzadeh, Mohammad},
	month = apr,
	year = {2017},
	note = {ISSN: 2640-3498},
	pages = {1468--1476},
}

@article{blanchard_hierarchical_2005,
	title = {Hierarchical testing designs for pattern recognition},
	volume = {33},
	issn = {0090-5364},
	url = {http://arxiv.org/abs/math/0507421},
	doi = {10.1214/009053605000000174},
	abstract = {We explore the theoretical foundations of a ``twenty questions'' approach to pattern recognition. The object of the analysis is the computational process itself rather than probability distributions (Bayesian inference) or decision boundaries (statistical learning). Our formulation is motivated by applications to scene interpretation in which there are a great many possible explanations for the data, one (``background'') is statistically dominant, and it is imperative to restrict intensive computation to genuinely ambiguous regions. The focus here is then on pattern filtering: Given a large set Y of possible patterns or explanations, narrow down the true one Y to a small (random) subset {\textbackslash}hat Y{\textbackslash}subsetY of ``detected'' patterns to be subjected to further, more intense, processing. To this end, we consider a family of hypothesis tests for Y{\textbackslash}in A versus the nonspecific alternatives Y{\textbackslash}in A{\textasciicircum}c. Each test has null type I error and the candidate sets A{\textbackslash}subsetY are arranged in a hierarchy of nested partitions. These tests are then characterized by scope ({\textbar}A{\textbar}), power (or type II error) and algorithmic cost. We consider sequential testing strategies in which decisions are made iteratively, based on past outcomes, about which test to perform next and when to stop testing. The set {\textbackslash}hat Y is then taken to be the set of patterns that have not been ruled out by the tests performed. The total cost of a strategy is the sum of the ``testing cost'' and the ``postprocessing cost'' (proportional to {\textbar}{\textbackslash}hat Y{\textbar}) and the corresponding optimization problem is analyzed.},
	number = {3},
	urldate = {2024-02-12},
	journal = {The Annals of Statistics},
	author = {Blanchard, Gilles and Geman, Donald},
	month = jun,
	year = {2005},
	note = {arXiv:math/0507421},
	keywords = {62H30{\textbackslash}sep62L05{\textbackslash}sep68T10 (Primary) 62H15{\textbackslash}sep68T45{\textbackslash}sep90B40 (Secondary), Mathematics - Statistics Theory},
}

@article{meinshausen_hierarchical_2008,
	title = {Hierarchical testing of variable importance},
	volume = {95},
	issn = {0006-3444, 1464-3510},
	url = {https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/asn007},
	doi = {10.1093/biomet/asn007},
	abstract = {A frequently encountered challenge in high-dimensional regression is the detection of relevant variables. Variable selection suﬀers from instability and the power to detect relevant variables is typically low if predictor variables are highly correlated. When taking the multiplicity of the testing problem into account, the power diminishes even further.},
	language = {en},
	number = {2},
	urldate = {2024-02-12},
	journal = {Biometrika},
	author = {Meinshausen, N.},
	month = feb,
	year = {2008},
	pages = {265--278},
}

@article{riedel_dynamically_2018,
	title = {Dynamically consistent preferences under imprecise probabilistic information},
	volume = {79},
	issn = {03044068},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0304406818300557},
	doi = {10.1016/j.jmateco.2018.04.006},
	abstract = {This paper extends decision theory under imprecise probabilistic information to dynamic settings. We explore the relationship between the given objective probabilistic information, an agent’s subjective multiple priors, and updating. Dynamic consistency implies rectangular sets of priors at the subjective level. As the objective probabilistic information need not be consistent with rectangularity at the subjective level, agents might select priors outside the objective probabilistic information while respecting the support of the given set of priors. Under suitable additional axioms, the subjective set of priors belongs to the rectangular hull of the objective probabilistic information.},
	language = {en},
	urldate = {2024-01-28},
	journal = {Journal of Mathematical Economics},
	author = {Riedel, Frank and Tallon, Jean-Marc and Vergopoulos, Vassili},
	month = dec,
	year = {2018},
	pages = {117--124},
}
@incollection{junqueira_barbosa_preference_2014,
	address = {Cham},
	title = {Preference {Relations} and {Families} of {Probabilities}: {Different} {Sides} of the {Same} {Coin}},
	volume = {442},
	isbn = {978-3-319-08794-8 978-3-319-08795-5},
	shorttitle = {Preference {Relations} and {Families} of {Probabilities}},
	url = {http://link.springer.com/10.1007/978-3-319-08795-5_1},
	urldate = {2024-01-28},
	booktitle = {Information {Processing} and {Management} of {Uncertainty} in {Knowledge}-{Based} {Systems}},
	publisher = {Springer International Publishing},
	author = {Couso, Inés},
	editor = {Junqueira Barbosa, Simone Diniz and Chen, Phoebe and Cuzzocrea, Alfredo and Du, Xiaoyong and Filipe, Joaquim and Kara, Orhun and Kotenko, Igor and Sivalingam, Krishna M. and Ślęzak, Dominik and Washio, Takashi and Yang, Xiaokang and Laurent, Anne and Strauss, Olivier and Bouchon-Meunier, Bernadette and Yager, Ronald R.},
	year = {2014},
	doi = {10.1007/978-3-319-08795-5_1},
	note = {Series Title: Communications in Computer and Information Science},
	pages = {1--9},
}

@book{noauthor_notitle_nodate,
}

@inproceedings{karimi_relationship_2023,
	title = {On the {Relationship} {Between} {Explanation} and {Prediction}: {A} {Causal} {View}},
	shorttitle = {On the {Relationship} {Between} {Explanation} and {Prediction}},
	url = {https://proceedings.mlr.press/v202/karimi23a.html},
	abstract = {Being able to provide explanations for a model’s decision has become a central requirement for the development, deployment, and adoption of machine learning models. However, we are yet to understand what explanation methods can and cannot do. How do upstream factors such as data, model prediction, hyperparameters, and random initialization influence downstream explanations? While previous work raised concerns that explanations (E) may have little relationship with the prediction (Y), there is a lack of conclusive study to quantify this relationship. Our work borrows tools from causal inference to systematically assay this relationship. More specifically, we study the relationship between E and Y by measuring the treatment effect when intervening on their causal ancestors, i.e., on hyperparameters and inputs used to generate saliency-based Es or Ys. Our results suggest that the relationships between E and Y is far from ideal. In fact, the gap between ’ideal’ case only increase in higher-performing models — models that are likely to be deployed. Our work is a promising first step towards providing a quantitative measure of the relationship between E and Y, which could also inform the future development of methods for E with a quantitative metric.},
	language = {en},
	urldate = {2024-01-26},
	booktitle = {Proceedings of the 40th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Karimi, Amir-Hossein and Muandet, Krikamol and Kornblith, Simon and Schölkopf, Bernhard and Kim, Been},
	month = jul,
	year = {2023},
	pages = {15861--15883},
}

@article{mendler-dunner_anticipating_2022,
	title = {Anticipating {Performativity} by {Predicting} from {Predictions}},
	volume = {35},
	url = {https://proceedings.neurips.cc/paper_files/paper/2022/hash/ca09b375e8e2b2c789698c079a9fc51c-Abstract-Conference.html},
	language = {en},
	urldate = {2024-01-26},
	journal = {Advances in Neural Information Processing Systems},
	author = {Mendler-Dünner, Celestine and Ding, Frances and Wang, Yixin},
	month = dec,
	year = {2022},
	pages = {31171--31185},
}

@inproceedings{horowitz_causal_2023,
	title = {Causal {Strategic} {Classification}: {A} {Tale} of {Two} {Shifts}},
	shorttitle = {Causal {Strategic} {Classification}},
	url = {https://proceedings.mlr.press/v202/horowitz23a.html},
	abstract = {When users can benefit from certain predictive outcomes, they may be prone to act to achieve those outcome, e.g., by strategically modifying their features. The goal in strategic classification is therefore to train predictive models that are robust to such behavior. However, the conventional framework assumes that changing features does not change actual outcomes, which depicts users as "gaming" the system. Here we remove this assumption, and study learning in a causal strategic setting where true outcomes do change. Focusing on accuracy as our primary objective, we show how strategic behavior and causal effects underlie two complementing forms of distribution shift. We characterize these shifts, and propose a learning algorithm that balances between these two forces and over time, and permits end-to-end training. Experiments on synthetic and semi-synthetic data demonstrate the utility of our approach.},
	language = {en},
	urldate = {2024-01-26},
	booktitle = {Proceedings of the 40th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Horowitz, Guy and Rosenfeld, Nir},
	month = jul,
	year = {2023},
	pages = {13233--13253},
}

@inproceedings{bechavod_information_2022,
	title = {Information {Discrepancy} in {Strategic} {Learning}},
	url = {https://proceedings.mlr.press/v162/bechavod22a.html},
	abstract = {We initiate the study of the effects of non-transparency in decision rules on individuals’ ability to improve in strategic learning settings. Inspired by real-life settings, such as loan approvals and college admissions, we remove the assumption typically made in the strategic learning literature, that the decision rule is fully known to individuals, and focus instead on settings where it is inaccessible. In their lack of knowledge, individuals try to infer this rule by learning from their peers (e.g., friends and acquaintances who previously applied for a loan), naturally forming groups in the population, each with possibly different type and level of information regarding the decision rule. We show that, in equilibrium, the principal’s decision rule optimizing welfare across sub-populations may cause a strong negative externality: the true quality of some of the groups can actually deteriorate. On the positive side, we show that, in many natural cases, optimal improvement can be guaranteed simultaneously for all sub-populations. We further introduce a measure we term information overlap proxy, and demonstrate its usefulness in characterizing the disparity in improvements across sub-populations. Finally, we identify a natural condition under which improvement can be guaranteed for all sub-populations while maintaining high predictive accuracy. We complement our theoretical analysis with experiments on real-world datasets.},
	language = {en},
	urldate = {2024-01-26},
	booktitle = {Proceedings of the 39th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Bechavod, Yahav and Podimata, Chara and Wu, Steven and Ziani, Juba},
	month = jun,
	year = {2022},
	pages = {1691--1715},
}

@inproceedings{ghalme_strategic_2021,
	title = {Strategic {Classification} in the {Dark}},
	url = {https://proceedings.mlr.press/v139/ghalme21a.html},
	abstract = {Strategic classification studies the interaction between a classification rule and the strategic agents it governs. Agents respond by manipulating their features, under the assumption that the classifier is known. However, in many real-life scenarios of high-stake classification (e.g., credit scoring), the classifier is not revealed to the agents, which leads agents to attempt to learn the classifier and game it too. In this paper we generalize the strategic classification model to such scenarios and analyze the effect of an unknown classifier. We define the ”price of opacity” as the difference between the prediction error under the opaque and transparent policies, characterize it, and give a sufficient condition for it to be strictly positive, in which case transparency is the recommended policy. Our experiments show how Hardt et al.’s robust classifier is affected by keeping agents in the dark.},
	language = {en},
	urldate = {2024-01-26},
	booktitle = {Proceedings of the 38th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Ghalme, Ganesh and Nair, Vineet and Eilat, Itay and Talgam-Cohen, Inbal and Rosenfeld, Nir},
	month = jul,
	year = {2021},
	pages = {3672--3681},
}

@article{shafer_tutorial_nodate,
	title = {A {Tutorial} on {Conformal} {Prediction}},
	abstract = {Conformal prediction uses past experience to determine precise levels of conﬁdence in new predictions. Given an error probability ε, together with a method that makes a prediction yˆ of a label y, it produces a set of labels, typically containing yˆ, that also contains y with probability 1 − ε. Conformal prediction can be applied to any method for producing yˆ: a nearest-neighbor method, a support-vector machine, ridge regression, etc.},
	language = {en},
	author = {Shafer, Glenn and Vovk, Vladimir},
}

@article{shafer_tutorial_nodate-1,
	title = {A {Tutorial} on {Conformal} {Prediction}},
	abstract = {Conformal prediction uses past experience to determine precise levels of conﬁdence in new predictions. Given an error probability ε, together with a method that makes a prediction yˆ of a label y, it produces a set of labels, typically containing yˆ, that also contains y with probability 1 − ε. Conformal prediction can be applied to any method for producing yˆ: a nearest-neighbor method, a support-vector machine, ridge regression, etc.},
	language = {en},
	author = {Shafer, Glenn and Vovk, Vladimir},
}

@misc{colin_what_2023,
	title = {What {I} {Cannot} {Predict}, {I} {Do} {Not} {Understand}: {A} {Human}-{Centered} {Evaluation} {Framework} for {Explainability} {Methods}},
	shorttitle = {What {I} {Cannot} {Predict}, {I} {Do} {Not} {Understand}},
	url = {http://arxiv.org/abs/2112.04417},
	abstract = {A multitude of explainability methods and associated fidelity performance metrics have been proposed to help better understand how modern AI systems make decisions. However, much of the current work has remained theoretical -- without much consideration for the human end-user. In particular, it is not yet known (1) how useful current explainability methods are in practice for more real-world scenarios and (2) how well associated performance metrics accurately predict how much knowledge individual explanations contribute to a human end-user trying to understand the inner-workings of the system. To fill this gap, we conducted psychophysics experiments at scale to evaluate the ability of human participants to leverage representative attribution methods for understanding the behavior of different image classifiers representing three real-world scenarios: identifying bias in an AI system, characterizing the visual strategy it uses for tasks that are too difficult for an untrained non-expert human observer as well as understanding its failure cases. Our results demonstrate that the degree to which individual attribution methods help human participants better understand an AI system varied widely across these scenarios. This suggests a critical need for the field to move past quantitative improvements of current attribution methods towards the development of complementary approaches that provide qualitatively different sources of information to human end-users.},
	urldate = {2024-01-23},
	publisher = {arXiv},
	author = {Colin, Julien and Fel, Thomas and Cadene, Remi and Serre, Thomas},
	month = jan,
	year = {2023},
	note = {arXiv:2112.04417 [cs]},
	keywords = {Computer Science - Computer Vision and Pattern Recognition, Computer Science - Human-Computer Interaction, Computer Science - Machine Learning},
}

@article{csaji_sign-perturbed_2015,
	title = {Sign-{Perturbed} {Sums}: {A} {New} {System} {Identification} {Approach} for {Constructing} {Exact} {Non}-{Asymptotic} {Confidence} {Regions} in {Linear} {Regression} {Models}},
	volume = {63},
	issn = {1053-587X, 1941-0476},
	shorttitle = {Sign-{Perturbed} {Sums}},
	url = {http://arxiv.org/abs/1807.08216},
	doi = {10.1109/TSP.2014.2369000},
	abstract = {We propose a new system identification method, called Sign-Perturbed Sums (SPS), for constructing non-asymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not limited to FIR systems, and we show that the SPS confidence regions have exact confidence probabilities, i.e., they contain the true parameter with a user-chosen exact probability for any finite data set. Moreover, we also prove that the SPS regions are star convex with the Least-Squares (LS) estimate as a star center. The main assumptions of SPS are that the noise terms are independent and symmetrically distributed about zero, but they can be nonstationary, and their distributions need not be known. The paper also proposes a computationally efficient ellipsoidal outer approximation algorithm for SPS. Finally, SPS is demonstrated through a number of simulation experiments.},
	number = {1},
	urldate = {2024-01-23},
	journal = {IEEE Transactions on Signal Processing},
	author = {Csáji, Balázs Cs and Campi, Marco C. and Weyer, Erik},
	month = jan,
	year = {2015},
	note = {arXiv:1807.08216 [cs, eess, stat]},
	keywords = {Computer Science - Machine Learning, Electrical Engineering and Systems Science - Signal Processing, Statistics - Methodology},
	pages = {169--181},
}

@misc{singh_kernel_2020,
	title = {Kernel {Instrumental} {Variable} {Regression}},
	url = {http://arxiv.org/abs/1906.00232},
	abstract = {Instrumental variable (IV) regression is a strategy for learning causal relationships in observational data. If measurements of input X and output Y are confounded, the causal relationship can nonetheless be identified if an instrumental variable Z is available that influences X directly, but is conditionally independent of Y given X and the unmeasured confounder. The classic two-stage least squares algorithm (2SLS) simplifies the estimation problem by modeling all relationships as linear functions. We propose kernel instrumental variable regression (KIV), a nonparametric generalization of 2SLS, modeling relations among X, Y, and Z as nonlinear functions in reproducing kernel Hilbert spaces (RKHSs). We prove the consistency of KIV under mild assumptions, and derive conditions under which convergence occurs at the minimax optimal rate for unconfounded, single-stage RKHS regression. In doing so, we obtain an efficient ratio between training sample sizes used in the algorithm's first and second stages. In experiments, KIV outperforms state of the art alternatives for nonparametric IV regression.},
	urldate = {2024-01-22},
	publisher = {arXiv},
	author = {Singh, Rahul and Sahani, Maneesh and Gretton, Arthur},
	month = jul,
	year = {2020},
	note = {arXiv:1906.00232 [cs, econ, math, stat]},
	keywords = {Computer Science - Machine Learning, Economics - Econometrics, Mathematics - Functional Analysis, Mathematics - Statistics Theory, Statistics - Machine Learning},
}

@misc{singh_indirect_2023,
	title = {Indirect {Active} {Learning}},
	url = {http://arxiv.org/abs/2206.01454},
	abstract = {Traditional models of active learning assume a learner can directly manipulate or query a covariate \$X\$ in order to study its relationship with a response \$Y\$. However, if \$X\$ is a feature of a complex system, it may be possible only to indirectly influence \$X\$ by manipulating a control variable \$Z\$, a scenario we refer to as Indirect Active Learning. Under a nonparametric model of Indirect Active Learning with a fixed budget, we study minimax convergence rates for estimating the relationship between \$X\$ and \$Y\$ locally at a point, obtaining different rates depending on the complexities and noise levels of the relationships between \$Z\$ and \$X\$ and between \$X\$ and \$Y\$. We also identify minimax rates for passive learning under comparable assumptions. In many cases, our results show that, while there is an asymptotic benefit to active learning, this benefit is fully realized by a simple two-stage learner that runs two passive experiments in sequence.},
	urldate = {2024-01-19},
	publisher = {arXiv},
	author = {Singh, Shashank},
	month = jan,
	year = {2023},
	note = {arXiv:2206.01454 [cs, math, stat]},
	keywords = {Computer Science - Information Theory, Computer Science - Machine Learning, Mathematics - Statistics Theory, Statistics - Machine Learning, Statistics - Methodology},
}

@article{csiszar_axiomatic_2008,
	title = {Axiomatic {Characterizations} of {Information} {Measures}},
	volume = {10},
	issn = {1099-4300},
	url = {http://www.mdpi.com/1099-4300/10/3/261},
	doi = {10.3390/e10030261},
	abstract = {Axiomatic characterizations of Shannon entropy, Kullback I-divergence, and some generalized information measures are surveyed. Three directions are treated: (A) Characterization of functions of probability distributions suitable as information measures. (B) Characterization of set functions on the subsets of \{1, . . . , N \} representable by joint entropies of components of an N -dimensional random vector. (C) Axiomatic characterization of MaxEnt and related inference rules. The paper concludes with a brief discussion of the relevance of the axiomatic approach for information theory.},
	language = {en},
	number = {3},
	urldate = {2024-01-19},
	journal = {Entropy},
	author = {Csiszár, Imre},
	month = sep,
	year = {2008},
	pages = {261--273},
}

@article{singh_nonparametric_nodate,
	title = {Nonparametric {Indirect} {Active} {Learning}},
	abstract = {Typical models of active learning assume a learner can directly manipulate or query a covariate X to study its relationship with a response Y . However, if X is a feature of a complex system, it may be possible only to indirectly influence X by manipulating a control variable Z, a scenario we refer to as Indirect Active Learning. Under a nonparametric fixed-budget model of Indirect Active Learning, we study minimax convergence rates for estimating a local relationship between X and Y , with different rates depending on the complexities and noise levels of the relationships between Z and X and between X and Y . We also derive minimax rates for passive learning under comparable assumptions, finding in many cases that, while there is an asymptotic benefit to active learning, this benefit is fully realized by a simple two-stage learner that runs two passive experiments in sequence.},
	language = {en},
	author = {Singh, Shashank},
}

@book{schulte_proceedings_2023,
	title = {Proceedings - 33. {Workshop} {Computational} {Intelligence}: {Berlin}, 23.-24. {November} 2023},
	isbn = {978-3-7315-1324-7},
	shorttitle = {Proceedings - 33. {Workshop} {Computational} {Intelligence}},
	url = {https://www.ksp.kit.edu/site/books/e/10.5445/KSP/1000162754/},
	abstract = {The workshop proceedings contain the contributions of the 33rd workshop "Computational Intelligence" which will take place from 23.11. - 24.11.2023 in Berlin. The focus is on methods, applications and tools for ° Fuzzy systems, ° Artificial Neural Networks, ° Evolutionary algorithms and ° Data mining methods as well as the comparison of methods on the basis of industrial and benchmark problems.},
	language = {en},
	urldate = {2024-01-19},
	publisher = {KIT Scientific Publishing},
	author = {Schulte, Horst},
	month = nov,
	year = {2023},
	doi = {10.58895/ksp/1000162754},
}

@inproceedings{le_heteroscedastic_2005,
	address = {Bonn, Germany},
	title = {Heteroscedastic {Gaussian} process regression},
	isbn = {978-1-59593-180-1},
	url = {http://portal.acm.org/citation.cfm?doid=1102351.1102413},
	doi = {10.1145/1102351.1102413},
	abstract = {This paper presents an algorithm to estimate simultaneously both mean and variance of a non parametric regression problem. The key point is that we are able to estimate variance locally unlike standard Gaussian Process regression or SVMs. This means that our estimator adapts to the local noise. The problem is cast in the setting of maximum a posteriori estimation in exponential families. Unlike previous work, we obtain a convex optimization problem which can be solved via Newton’s method.},
	language = {en},
	urldate = {2024-01-19},
	booktitle = {Proceedings of the 22nd international conference on {Machine} learning  - {ICML} '05},
	publisher = {ACM Press},
	author = {Le, Quoc V. and Smola, Alex J. and Canu, Stéphane},
	year = {2005},
	pages = {489--496},
}

@article{abellan_measures_2006,
	title = {Measures of divergence on credal sets},
	volume = {157},
	issn = {01650114},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0165011405005932},
	doi = {10.1016/j.fss.2005.11.021},
	abstract = {When there is uncertainty about the values of a variable, it is possible to have different pieces of information about it. This can be due to the fact that they represent the points of view of different agents and these agents have followed different criteria when codify the pieces of information by means of numbers. The aim of this paper is to provide tools for the comparison of these representations. This is done in the framework of imprecise probabilities, more concretely the credal set theory is used. The reason is that credal sets are very general, including as particular cases possibility theory, probability theory and belief functions. So it can serve as basis to compare the representations of information obtained by following these formalisms. The measures used for comparison are: inconsistency measure, inclusion index and informative distance.},
	language = {en},
	number = {11},
	urldate = {2024-01-13},
	journal = {Fuzzy Sets and Systems},
	author = {Abellán, Joaquín and Gómez, Manuel},
	month = jun,
	year = {2006},
	pages = {1514--1531},
}

@article{destercke_handling_2012,
	title = {Handling bipolar knowledge with imprecise probabilities},
	volume = {26},
	url = {https://hal.science/hal-00652236},
	doi = {10.1002/int.20475},
	abstract = {Information is said to be bipolar when it has a positive and a negative part. The problem of representing and processing such bipolar information has recently received a lot of attention in uncertainty theories. In this paper, we are concerned with the representation of asymmetric bipolarity, i.e., with situations where positive and negative information are unrelated and processed in parallel. In this latter case, positive information consists of observations of experiment results, showing what values are possible, whereas negative information consists of constraints (e.g., provided by an expert), restricting the range of possible variable values. Up to now, there are no proposition as to how such bipolar information can be treated in the framework of imprecise probability theory, i.e., when information is represented by convex sets of probabilities. In this paper, we propose the basis of such a framework and provide some illustrative examples.},
	number = {5},
	urldate = {2024-01-13},
	journal = {International Journal of Intelligent Systems},
	author = {Destercke, Sébastien},
	month = mar,
	year = {2012},
	note = {Publisher: Wiley},
	keywords = {BIPOLARITE ASYMETRIQUE, BIPOLARITY, CREDAL SETS, INFORMATION BIPOLAIRE, INFORMATION FUSION, REPRESENTATION DES INCERTITUDES, THEORIE DES PROBABILITES IMPRECISES, UNCERTAINTY REPRESENTATION, VALEURS DES VARIABLES},
	pages = {426--443},
}

@article{veazie_when_2006,
	title = {When to {Combine} {Hypotheses} and {Adjust} for {Multiple} {Tests}},
	volume = {41},
	issn = {0017-9124, 1475-6773},
	url = {https://onlinelibrary.wiley.com/doi/10.1111/j.1475-6773.2006.00512.x},
	doi = {10.1111/j.1475-6773.2006.00512.x},
	abstract = {Objective. To provide guidelines for identifying composite hypotheses and addressing the probability of false rejection for multiple hypotheses. Data Sources and Study Setting. Examples from the literature in health services research are used to motivate the discussion of composite hypothesis tests and multiple hypotheses.
Methods. This article is a didactic presentation. Principal Findings. It is not rare to ﬁnd mistaken inferences in health services research because of inattention to appropriate hypothesis generation and multiple hypotheses testing. Guidelines are presented to help researchers identify composite hypotheses and set significance levels to account for multiple tests.
Conclusions. It is important for the quality of scholarship that inferences are valid: properly identifying composite hypotheses and accounting for multiple tests provides some assurance in this regard.},
	language = {en},
	number = {3p1},
	urldate = {2024-01-12},
	journal = {Health Services Research},
	author = {Veazie, Peter J.},
	month = jun,
	year = {2006},
	pages = {804--818},
}

@article{key_composite_nodate,
	title = {Composite {Goodness}-of-fit {Tests} with {Kernels}},
	abstract = {Model misspecification can create significant challenges for the implementation of probabilistic models, and this has led to development of a range of robust methods which directly account for this issue. However, whether these more involved methods are required will depend on whether the model is really misspecified, and there is a lack of generally applicable methods to answer this question. In this paper, we propose one such method. More precisely, we propose kernel-based hypothesis tests for the challenging composite testing problem, where we are interested in whether the data comes from any distribution in some parametric family. Our tests make use of minimum distance estimators based on the maximum mean discrepancy and the kernel Stein discrepancy. They are widely applicable, including whenever the density of the parametric model is known up to normalisation constant, or if the model takes the form of a simulator. As our main result, we show that we are able to estimate the parameter and conduct our test on the same data (without data splitting), while maintaining a correct test level. Our approach is illustrated on a range of problems, including testing for goodness-of-fit of an unnormalised non-parametric density model, and an intractable generative model of a biological cellular network.},
	language = {en},
	author = {Key, Oscar},
}

@article{huang_kernel_2023,
	title = {A {Kernel} {Measure} of {Dissimilarity} between \textit{{M}} {Distributions}},
	issn = {0162-1459, 1537-274X},
	url = {https://www.tandfonline.com/doi/full/10.1080/01621459.2023.2298036},
	doi = {10.1080/01621459.2023.2298036},
	language = {en},
	urldate = {2024-01-11},
	journal = {Journal of the American Statistical Association},
	author = {Huang, Zhen and Sen, Bodhisattva},
	month = dec,
	year = {2023},
	pages = {1--26},
}

@incollection{sidoravicius_distribution_2002,
	address = {Boston, MA},
	title = {The {Distribution} of the {Maximum} of a {Gaussian} {Process}: {Rice} {Method} {Revisited}},
	isbn = {978-1-4612-6595-5 978-1-4612-0063-5},
	shorttitle = {The {Distribution} of the {Maximum} of a {Gaussian} {Process}},
	url = {https://link.springer.com/10.1007/978-1-4612-0063-5_15},
	abstract = {This paper deals with the problem of obtaining methods to compute the distribution of the maximum of a one-parameter stochastic process on a ﬁxed interval, mainly in the Gaussian case. The main point is the relationship between the values of the maximum and crossings of the paths, via the so-called Rice’s formulae for the factorial moments of crossings.},
	language = {en},
	urldate = {2024-01-10},
	booktitle = {In and {Out} of {Equilibrium}},
	publisher = {Birkhäuser Boston},
	author = {Azaïs, Jean-Marc and Wschebor, Mario},
	editor = {Sidoravicius, Vladas},
	year = {2002},
	doi = {10.1007/978-1-4612-0063-5_15},
	pages = {321--348},
}

@misc{bellot_kernel_2021,
	title = {Kernel {Hypothesis} {Testing} with {Set}-valued {Data}},
	url = {http://arxiv.org/abs/1907.04081},
	abstract = {We present a general framework for hypothesis testing on distributions of sets of individual examples. Sets may represent many common data sources such as groups of observations in time series, collections of words in text or a batch of images of a given phenomenon. This observation pattern, however, differs from the common assumptions required for hypothesis testing: each set differs in size, may have differing levels of noise, and also may incorporate nuisance variability, irrelevant for the analysis of the phenomenon of interest; all features that bias test decisions if not accounted for. In this paper, we propose to interpret sets as independent samples from a collection of latent probability distributions, and introduce kernel two-sample and independence tests in this latent space of distributions. We prove the consistency of tests and observe them to outperform in a wide range of synthetic experiments. Finally, we showcase their use in practice with experiments of healthcare and climate data, where previously heuristics were needed for feature extraction and testing.},
	urldate = {2024-01-10},
	publisher = {arXiv},
	author = {Bellot, Alexis and van der Schaar, Mihaela},
	month = feb,
	year = {2021},
	note = {arXiv:1907.04081 [stat]},
	keywords = {Statistics - Machine Learning, Statistics - Methodology},
}

@article{sun_kernel_nodate,
	title = {Kernel {Robust} {Hypothesis} {Testing}},
	abstract = {The problem of robust hypothesis testing is studied, where under the null and the alternative hypotheses, the data-generating distributions are assumed to be in some uncertainty sets, and the goal is to design a test that performs well under the worst-case distributions over the uncertainty sets. In this paper, uncertainty sets are constructed in a data-driven manner using kernel method, i.e., they are centered around empirical distributions of training samples from the null and alternative hypotheses, respectively; and are constrained via the distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space, i.e., maximum mean discrepancy (MMD). The Bayesian setting and the NeymanPearson setting are investigated. For the Bayesian setting where the goal is to minimize the worst-case error probability, an optimal test is firstly obtained when the alphabet is finite. When the alphabet is infinite, a tractable approximation is proposed to quantify the worst-case average error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A direct robust kernel test is also proposed and proved to be exponentially consistent. For the Neyman-Pearson setting, where the goal is to minimize the worst-case probability of miss detection subject to a constraint on the worst-case probability of false alarm, an efficient robust kernel test is proposed and is shown to be asymptotically optimal. Numerical results are provided to demonstrate the performance of the proposed robust tests.},
	language = {en},
	author = {Sun, Zhongchang and Zou, Shaofeng},
}

@article{xu_advances_nodate,
	title = {Advances in {Non}-parametric {Hypothesis} {Testing} with {Kernels}},
	language = {en},
	author = {Xu, Wenkai},
}

@article{zhang_optimal_2020,
	title = {Optimal {Transport} in {Reproducing} {Kernel} {Hilbert} {Spaces}: {Theory} and {Applications}},
	volume = {42},
	issn = {0162-8828, 2160-9292, 1939-3539},
	shorttitle = {Optimal {Transport} in {Reproducing} {Kernel} {Hilbert} {Spaces}},
	url = {https://ieeexplore.ieee.org/document/8658140/},
	doi = {10.1109/TPAMI.2019.2903050},
	abstract = {In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel Hilbert spaces (RKHS). This framework, called optimal transport in RKHS, is a generalization of the optimal transport problem in input spaces to (potentially) inﬁnite-dimensional feature spaces. We provide a computable formulation of Kantorovich’s optimal transport in RKHS. In particular, we explore the case in which data distributions in RKHS are Gaussian, obtaining closed-form expressions of both the estimated Wasserstein distance and optimal transport map via kernel matrices. Based on these expressions, we generalize the Bures metric on covariance matrices to inﬁnite-dimensional settings, providing a new metric between covariance operators. Moreover, we extend the correlation alignment problem to Hilbert spaces, giving a new strategy for matching distributions in RKHS. Empirically, we apply the derived formulas under the Gaussianity assumption to image classiﬁcation and domain adaptation. In both tasks, our algorithms yield state-of-the-art performances, demonstrating the effectiveness and potential of our framework.},
	language = {en},
	number = {7},
	urldate = {2024-01-03},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	author = {Zhang, Zhen and Wang, Mianzhi and Nehorai, Arye},
	month = jul,
	year = {2020},
	pages = {1741--1754},
}

@article{kaufmann_survey_nodate,
	title = {A {Survey} of {Reinforcement} {Learning} from {Human} {Feedback}},
	abstract = {Reinforcement learning from human feedback (RLHF) is a variant of reinforcement learning (RL) that learns from human feedback instead of relying on an engineered reward function. Building on prior work on the related setting of preference-based reinforcement learning (PbRL), it stands at the intersection of artificial intelligence and human-computer interaction. This positioning offers a promising avenue to enhance the performance and adaptability of intelligent systems while also improving the alignment of their objectives with human values. The training of large language models (LLMs) has impressively demonstrated this potential in recent years, where RLHF played a decisive role in targeting the model’s capabilities toward human objectives. This article provides a comprehensive overview of the fundamentals of RLHF, exploring the intricate dynamics between machine agents and human input. While recent focus has been on RLHF for LLMs, our survey adopts a broader perspective, examining the diverse applications and wide-ranging impact of the technique. We delve into the core principles that underpin RLHF, shedding light on the symbiotic relationship between algorithms and human feedback, and discuss the main research trends in the field. By synthesizing the current landscape of RLHF research, this article aims to provide researchers as well as practitioners with a comprehensive understanding of this rapidly growing field of research.},
	language = {en},
	author = {Kaufmann, Timo and Weng, Paul and Bengs, Viktor and Hüllermeier, Eyke},
}

@inproceedings{jia_proof--learning_2021,
	address = {San Francisco, CA, USA},
	title = {Proof-of-{Learning}: {Definitions} and {Practice}},
	isbn = {978-1-72818-934-5},
	shorttitle = {Proof-of-{Learning}},
	url = {https://ieeexplore.ieee.org/document/9519402/},
	doi = {10.1109/SP40001.2021.00106},
	abstract = {Training machine learning (ML) models typically involves expensive iterative optimization. Once the model’s ﬁnal parameters are released, there is currently no mechanism for the entity which trained the model to prove that these parameters were indeed the result of this optimization procedure. Such a mechanism would support security of ML applications in several ways. For instance, it would simplify ownership resolution when multiple parties contest ownership of a speciﬁc model. It would also facilitate the distributed training across untrusted workers where Byzantine workers might otherwise mount a denial-ofservice by returning incorrect model updates.},
	language = {en},
	urldate = {2023-12-27},
	booktitle = {2021 {IEEE} {Symposium} on {Security} and {Privacy} ({SP})},
	publisher = {IEEE},
	author = {Jia, Hengrui and Yaghini, Mohammad and Choquette-Choo, Christopher A. and Dullerud, Natalie and Thudi, Anvith and Chandrasekaran, Varun and Papernot, Nicolas},
	month = may,
	year = {2021},
	pages = {1039--1056},
}

@misc{geshkovski_mathematical_2023,
	title = {A mathematical perspective on {Transformers}},
	url = {http://arxiv.org/abs/2312.10794},
	abstract = {Transformers play a central role in the inner workings of large language models. We develop a mathematical framework for analyzing Transformers based on their interpretation as interacting particle systems, which reveals that clusters emerge in long time. Our study explores the underlying theory and offers new perspectives for mathematicians as well as computer scientists.},
	urldate = {2023-12-21},
	publisher = {arXiv},
	author = {Geshkovski, Borjan and Letrouit, Cyril and Polyanskiy, Yury and Rigollet, Philippe},
	month = dec,
	year = {2023},
	note = {arXiv:2312.10794 [cs, math]},
	keywords = {Computer Science - Machine Learning, Mathematics - Analysis of PDEs, Mathematics - Dynamical Systems},
}

@misc{frieder_large_2023,
	title = {Large {Language} {Models} for {Mathematicians}},
	url = {http://arxiv.org/abs/2312.04556},
	abstract = {Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code. For many professions, LLMs represent an invaluable tool that can speed up and improve the quality of work. In this note, we discuss to what extent they can aid professional mathematicians. We first provide a mathematical description of the transformer model used in all modern language models. Based on recent studies, we then outline best practices and potential issues and report on the mathematical abilities of language models. Finally, we shed light on the potential of LMMs to change how mathematicians work.},
	urldate = {2023-12-21},
	publisher = {arXiv},
	author = {Frieder, Simon and Berner, Julius and Petersen, Philipp and Lukasiewicz, Thomas},
	month = dec,
	year = {2023},
	note = {arXiv:2312.04556 [cs, math]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Computation and Language, Computer Science - Machine Learning, Mathematics - History and Overview},
}

@article{cheney_causality_2010,
	title = {Causality and the {Semantics} of {Provenance}},
	volume = {26},
	issn = {2075-2180},
	url = {http://arxiv.org/abs/1006.1429v1},
	doi = {10.4204/EPTCS.26.6},
	language = {en},
	urldate = {2023-12-20},
	journal = {Electronic Proceedings in Theoretical Computer Science},
	author = {Cheney, James},
	month = jun,
	year = {2010},
	pages = {63--74},
}

@misc{wilson_deep_2015,
	title = {Deep {Kernel} {Learning}},
	url = {http://arxiv.org/abs/1511.02222},
	abstract = {We introduce scalable deep kernels, which combine the structural properties of deep learning architectures with the non-parametric flexibility of kernel methods. Specifically, we transform the inputs of a spectral mixture base kernel with a deep architecture, using local kernel interpolation, inducing points, and structure exploiting (Kronecker and Toeplitz) algebra for a scalable kernel representation. These closed-form kernels can be used as drop-in replacements for standard kernels, with benefits in expressive power and scalability. We jointly learn the properties of these kernels through the marginal likelihood of a Gaussian process. Inference and learning cost \$O(n)\$ for \$n\$ training points, and predictions cost \$O(1)\$ per test point. On a large and diverse collection of applications, including a dataset with 2 million examples, we show improved performance over scalable Gaussian processes with flexible kernel learning models, and stand-alone deep architectures.},
	urldate = {2023-12-20},
	publisher = {arXiv},
	author = {Wilson, Andrew Gordon and Hu, Zhiting and Salakhutdinov, Ruslan and Xing, Eric P.},
	month = nov,
	year = {2015},
	note = {arXiv:1511.02222 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning, Statistics - Methodology},
}

@article{wilson_pathwise_nodate,
	title = {Pathwise {Conditioning} of {Gaussian} {Processes}},
	abstract = {As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at ﬁnite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a diﬀerent line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to eﬃciently sampling Gaussian process posteriors. Starting from ﬁrst principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning.},
	language = {en},
	author = {Wilson, James T and Borovitskiy, Viacheslav and Terenin, Alexander and Mostowsky, Peter and Deisenroth, Marc Peter},
}

@misc{bordt_shapley_2023,
	title = {From {Shapley} {Values} to {Generalized} {Additive} {Models} and back},
	url = {http://arxiv.org/abs/2209.04012},
	abstract = {In explainable machine learning, local post-hoc explanation algorithms and inherently interpretable models are often seen as competing approaches. This work offers a partial reconciliation between the two by establishing a correspondence between Shapley Values and Generalized Additive Models (GAMs). We introduce \$n\$-Shapley Values, a parametric family of local post-hoc explanation algorithms that explain individual predictions with interaction terms up to order \$n\$. By varying the parameter \$n\$, we obtain a sequence of explanations that covers the entire range from Shapley Values up to a uniquely determined decomposition of the function we want to explain. The relationship between \$n\$-Shapley Values and this decomposition offers a functionally-grounded characterization of Shapley Values, which highlights their limitations. We then show that \$n\$-Shapley Values, as well as the Shapley Taylor- and Faith-Shap interaction indices, recover GAMs with interaction terms up to order \$n\$. This implies that the original Shapely Values recover GAMs without variable interactions. Taken together, our results provide a precise characterization of Shapley Values as they are being used in explainable machine learning. They also offer a principled interpretation of partial dependence plots of Shapley Values in terms of the underlying functional decomposition. A package for the estimation of different interaction indices is available at {\textbackslash}url\{https://github.com/tml-tuebingen/nshap\}.},
	urldate = {2023-12-16},
	publisher = {arXiv},
	author = {Bordt, Sebastian and von Luxburg, Ulrike},
	month = feb,
	year = {2023},
	note = {arXiv:2209.04012 [cs]},
	keywords = {Computer Science - Machine Learning},
}

@inproceedings{molinari_trust_2023,
	title = {Trust the evidence: two deference principles for imprecise probabilities},
	shorttitle = {Trust the evidence},
	url = {https://proceedings.mlr.press/v215/molinari23a.html},
	abstract = {Our intuition that rational agents should value the evidence can be captured by a well-known theorem due to I. J. Good. However, Good’s theorem fails when agents have imprecise credences, raising the worry that agents with imprecise credences don’t value the evidence. This essay shows a different way to capture our starting intuition, as the claim that rational agents defer to their informed selves. I introduce and motivate two deference principles for imprecise probabilities, and show that rational imprecise agents defer to their informed selves according to these principles. This shows a sense in which imprecise agents value the evidence. I end by comparing the deference principles introduced here with an alternative from the literature.},
	language = {en},
	urldate = {2023-12-15},
	booktitle = {Proceedings of the {Thirteenth} {International} {Symposium} on {Imprecise} {Probability}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Molinari, Giacomo},
	month = jul,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {356--366},
}

@article{csaji_distribution-free_2019,
	title = {Distribution-free uncertainty quantification for kernel methods by gradient perturbations},
	volume = {108},
	issn = {0885-6125, 1573-0565},
	url = {http://link.springer.com/10.1007/s10994-019-05822-1},
	doi = {10.1007/s10994-019-05822-1},
	abstract = {We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from ﬁnite-sample system identiﬁcation, that by perturbing the residuals in the gradient of the objective function, information can be extracted about the amount of uncertainty our model has. Particularly, we provide an algorithm to build exact, non-asymptotically guaranteed, distribution-free conﬁdence regions for ideal, noise-free representations of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efﬁcient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, ε-SVR and kernelized LASSO.},
	language = {en},
	number = {8-9},
	urldate = {2023-12-15},
	journal = {Machine Learning},
	author = {Csáji, Balázs Cs. and Kis, Krisztián B.},
	month = sep,
	year = {2019},
	pages = {1677--1699},
}

@misc{turner_introduction_2023,
	title = {An {Introduction} to {Transformers}},
	url = {http://arxiv.org/abs/2304.10557},
	abstract = {The transformer is a neural network component that can be used to learn useful representations of sequences or sets of data-points. The transformer has driven recent advances in natural language processing, computer vision, and spatio-temporal modelling. There are many introductions to transformers, but most do not contain precise mathematical descriptions of the architecture and the intuitions behind the design choices are often also missing. Moreover, as research takes a winding path, the explanations for the components of the transformer can be idiosyncratic. In this note we aim for a mathematically precise, intuitive, and clean description of the transformer architecture. We will not discuss training as this is rather standard. We assume that the reader is familiar with fundamental topics in machine learning including multi-layer perceptrons, linear transformations, softmax functions and basic probability.},
	urldate = {2023-12-14},
	publisher = {arXiv},
	author = {Turner, Richard E.},
	month = oct,
	year = {2023},
	note = {arXiv:2304.10557 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning},
}

@misc{pernot_calibration_2023,
	title = {Calibration in {Machine} {Learning} {Uncertainty} {Quantification}: beyond consistency to target adaptivity},
	shorttitle = {Calibration in {Machine} {Learning} {Uncertainty} {Quantification}},
	url = {http://arxiv.org/abs/2309.06240},
	abstract = {Reliable uncertainty quantification (UQ) in machine learning (ML) regression tasks is becoming the focus of many studies in materials and chemical science. It is now well understood that average calibration is insufficient, and most studies implement additional methods testing the conditional calibration with respect to uncertainty, i.e. consistency. Consistency is assessed mostly by so-called reliability diagrams. There exists however another way beyond average calibration, which is conditional calibration with respect to input features, i.e. adaptivity. In practice, adaptivity is the main concern of the final users of a ML-UQ method, seeking for the reliability of predictions and uncertainties for any point in features space. This article aims to show that consistency and adaptivity are complementary validation targets, and that a good consistency does not imply a good adaptivity. Adapted validation methods are proposed and illustrated on a representative example.},
	urldate = {2023-12-08},
	publisher = {arXiv},
	author = {Pernot, Pascal},
	month = dec,
	year = {2023},
	note = {arXiv:2309.06240 [physics, stat]},
	keywords = {Computer Science - Machine Learning, Physics - Chemical Physics, Physics - Data Analysis, Statistics and Probability, Statistics - Machine Learning},
}

@article{bonnier_kernelized_nodate,
	title = {Kernelized {Cumulants}: {Beyond} {Kernel} {Mean} {Embeddings}},
	abstract = {In Rd, it is well-known that cumulants provide an alternative to moments that can achieve the same goals with numerous benefits such as lower variance estimators. In this paper we extend cumulants to reproducing kernel Hilbert spaces (RKHS) using tools from tensor algebras and show that they are computationally tractable by a kernel trick. These kernelized cumulants provide a new set of all-purpose statistics; the classical maximum mean discrepancy and Hilbert-Schmidt independence criterion arise as the degree one objects in our general construction. We argue both theoretically and empirically (on synthetic, environmental, and traffic data analysis) that going beyond degree one has several advantages and can be achieved with the same computational complexity and minimal overhead in our experiments.},
	language = {en},
	author = {Bonnier, Patric and Oberhauser, Harald and Szabó, Zoltán},
}

@misc{sale_second-order_2023,
	title = {Second-{Order} {Uncertainty} {Quantification}: {A} {Distance}-{Based} {Approach}},
	shorttitle = {Second-{Order} {Uncertainty} {Quantification}},
	url = {http://arxiv.org/abs/2312.00995},
	abstract = {In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.},
	urldate = {2023-12-07},
	publisher = {arXiv},
	author = {Sale, Yusuf and Bengs, Viktor and Caprio, Michele and Hüllermeier, Eyke},
	month = dec,
	year = {2023},
	note = {arXiv:2312.00995 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{sale_second-order_2023-1,
	title = {Second-{Order} {Uncertainty} {Quantification}: {A} {Distance}-{Based} {Approach}},
	shorttitle = {Second-{Order} {Uncertainty} {Quantification}},
	url = {http://arxiv.org/abs/2312.00995},
	abstract = {In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.},
	urldate = {2023-12-07},
	publisher = {arXiv},
	author = {Sale, Yusuf and Bengs, Viktor and Caprio, Michele and Hüllermeier, Eyke},
	month = dec,
	year = {2023},
	note = {arXiv:2312.00995 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{de_campos_independence_2013,
	title = {Independence {Concepts} for {Convex} {Sets} of {Probabilities}},
	url = {http://arxiv.org/abs/1302.4940},
	abstract = {In this paper we study different concepts of independence for convex sets of probabilities. There will be two basic ideas for independence. The first is irrelevance. Two variables are independent when a change on the knowledge about one variable does not affect the other. The second one is factorization. Two variables are independent when the joint convex set of probabilities can be decomposed on the product of marginal convex sets. In the case of the Theory of Probability, these two starting points give rise to the same definition. In the case of convex sets of probabilities, the resulting concepts will be strongly related, but they will not be equivalent. As application of the concept of independence, we shall consider the problem of building a global convex set from marginal convex sets of probabilities.},
	urldate = {2023-12-06},
	publisher = {arXiv},
	author = {de Campos, Luis M. and Moral, Serafin},
	month = feb,
	year = {2013},
	note = {arXiv:1302.4940 [cs]},
	keywords = {Computer Science - Artificial Intelligence},
}

@misc{sagawa_distributionally_2020,
	title = {Distributionally {Robust} {Neural} {Networks} for {Group} {Shifts}: {On} the {Importance} of {Regularization} for {Worst}-{Case} {Generalization}},
	shorttitle = {Distributionally {Robust} {Neural} {Networks} for {Group} {Shifts}},
	url = {http://arxiv.org/abs/1911.08731},
	abstract = {Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-deﬁned groups. However, we ﬁnd that naively applying group DRO to overparameterized neural networks fails: these models can perfectly ﬁt the training data, and any model with vanishing average training loss also already has vanishing worst-case training loss. Instead, the poor worst-case performance arises from poor generalization on some groups. By coupling group DRO models with increased regularization—a stronger-than-typical 2 penalty or early stopping—we achieve substantially higher worst-group accuracies, with 10–40 percentage point improvements on a natural language inference task and two image tasks, while maintaining high average accuracies. Our results suggest that regularization is important for worst-group generalization in the overparameterized regime, even if it is not needed for average generalization. Finally, we introduce a stochastic optimization algorithm, with convergence guarantees, to efﬁciently train group DRO models.},
	language = {en},
	urldate = {2023-12-06},
	publisher = {arXiv},
	author = {Sagawa, Shiori and Koh, Pang Wei and Hashimoto, Tatsunori B. and Liang, Percy},
	month = apr,
	year = {2020},
	note = {arXiv:1911.08731 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{krueger_out--distribution_2021,
	title = {Out-of-{Distribution} {Generalization} via {Risk} {Extrapolation} ({REx})},
	url = {http://arxiv.org/abs/2003.00688},
	abstract = {Distributional shift is one of the major obstacles when transferring machine learning prediction systems from the lab to the real world. To tackle this problem, we assume that variation across training domains is representative of the variation we might encounter at test time, but also that shifts at test time may be more extreme in magnitude. In particular, we show that reducing differences in risk across training domains can reduce a model’s sensitivity to a wide range of extreme distributional shifts, including the challenging setting where the input contains both causal and anticausal elements. We motivate this approach, Risk Extrapolation (REx), as a form of robust optimization over a perturbation set of extrapolated domains (MM-REx), and propose a penalty on the variance of training risks (V-REx) as a simpler variant. We prove that variants of REx can recover the causal mechanisms of the targets, while also providing some robustness to changes in the input distribution (“covariate shift”). By tradingoff robustness to causally induced distributional shifts and covariate shift, REx is able to outperform alternative methods such as Invariant Risk Minimization in situations where these types of shift co-occur.},
	language = {en},
	urldate = {2023-12-06},
	publisher = {arXiv},
	author = {Krueger, David and Caballero, Ethan and Jacobsen, Joern-Henrik and Zhang, Amy and Binas, Jonathan and Zhang, Dinghuai and Priol, Remi Le and Courville, Aaron},
	month = feb,
	year = {2021},
	note = {arXiv:2003.00688 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing, Statistics - Machine Learning},
}

@article{mehra_domain_nodate,
	title = {Do {Domain} {Generalization} {Methods} {Generalize} {Well}?},
	abstract = {Domain Generalization (DG) methods use data from multiple related source domains to learn models whose performance does not degrade on unseen domains at test time. Many DG algorithms rely on reducing the divergence between the source distributions in a representation space to potentially align unseen domains close to the sources. These algorithms are motivated by the analytical works that explain generalization to unseen domains based on their distributional distance (e.g., Wasserstein distance) to the sources. However, we show that the accuracy of a DG model varies significantly on unseen domains equidistant from the sources in the learned representation space. This makes it hard to gauge the generalization performance of DG models only based on their performance on benchmark datasets. Thus, we study the worst-case loss of a DG model at a particular distance from the sources and propose an evaluation methodology based on distributionally robust optimization that efficiently computes the worst-case loss on all distributions within a Wasserstein ball around the sources. Our results show that models trained with popular DG methods incur a high worst-case loss even close to the sources which show their lack of generalization to unseen domains. Moreover, we observe a large gap between the worst-case and the empirical losses of distributions at the same distance, showing the performance of the DG models on benchmark datasets is not representative of their performance on unseen domains. Thus, our (target) data-independent and worst-case loss-based methodology highlights the poor generalization performance of current DG models and provides insights beyond empirical evaluation on benchmark datasets for improving these models.},
	language = {en},
	author = {Mehra, Akshay and Kailkhura, Bhavya and Chen, Pin-Yu and Hamm, Jihun},
}

@inproceedings{magesh_principled_2023,
	address = {Taipei, Taiwan},
	title = {Principled {OOD} {Detection} via {Multiple} {Testing}},
	isbn = {978-1-66547-554-9},
	url = {https://ieeexplore.ieee.org/document/10206581/},
	doi = {10.1109/ISIT54713.2023.10206581},
	abstract = {We study the problem of Out-of-Distribution (OOD) detection, that is, detecting whether a Machine Learning (ML) model’s output can be trusted at inference time. While a number of tests for OOD detection have been proposed in prior work, a formal framework for studying this problem is lacking. We propose a definition for the notion of OOD that includes both the input distribution and the ML model, which provides insights for the construction of powerful tests for OOD detection. We also propose a multiple hypothesis testing inspired procedure to systematically combine any number of different statistics from the ML model using conformal p-values. We further provide strong guarantees on the probability of incorrectly classifying an in-distribution sample as OOD. In our experiments, we find that threshold-based tests proposed in prior work perform well in specific settings, but not uniformly well across different OOD instances. In contrast, our proposed method that combines multiple statistics performs uniformly well across different datasets and neural networks.},
	language = {en},
	urldate = {2023-12-06},
	booktitle = {2023 {IEEE} {International} {Symposium} on {Information} {Theory} ({ISIT})},
	publisher = {IEEE},
	author = {Magesh, Akshayaa and Veeravalli, Venugopal V. and Roy, Anirban and Jha, Susmit},
	month = jun,
	year = {2023},
	pages = {1026--1031},
}

@misc{haroush_statistical_2022,
	title = {A statistical framework for efficient out of distribution detection in deep neural networks},
	url = {http://arxiv.org/abs/2102.12967},
	abstract = {Background. Commonly, Deep Neural Networks (DNNs) generalize well on samples drawn from a distribution similar to that of the training set. However, DNNs' predictions are brittle and unreliable when the test samples are drawn from a dissimilar distribution. This is a major concern for deployment in real-world applications, where such behavior may come at a considerable cost, such as industrial production lines, autonomous vehicles, or healthcare applications. Contributions. We frame Out Of Distribution (OOD) detection in DNNs as a statistical hypothesis testing problem. Tests generated within our proposed framework combine evidence from the entire network. Unlike previous OOD detection heuristics, this framework returns a \$p\$-value for each test sample. It is guaranteed to maintain the Type I Error (T1E - incorrectly predicting OOD for an actual in-distribution sample) for test data. Moreover, this allows to combine several detectors while maintaining the T1E. Building on this framework, we suggest a novel OOD procedure based on low-order statistics. Our method achieves comparable or better results than state-of-the-art methods on well-accepted OOD benchmarks, without retraining the network parameters or assuming prior knowledge on the test distribution -- and at a fraction of the computational cost.},
	urldate = {2023-12-06},
	publisher = {arXiv},
	author = {Haroush, Matan and Frostig, Tzviel and Heller, Ruth and Soudry, Daniel},
	month = mar,
	year = {2022},
	note = {arXiv:2102.12967 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{haroush_statistical_2022,
	title = {A statistical framework for efficient out of distribution detection in deep neural networks},
	url = {http://arxiv.org/abs/2102.12967},
	abstract = {Background. Commonly, Deep Neural Networks (DNNs) generalize well on samples drawn from a distribution similar to that of the training set. However, DNNs' predictions are brittle and unreliable when the test samples are drawn from a dissimilar distribution. This is a major concern for deployment in real-world applications, where such behavior may come at a considerable cost, such as industrial production lines, autonomous vehicles, or healthcare applications. Contributions. We frame Out Of Distribution (OOD) detection in DNNs as a statistical hypothesis testing problem. Tests generated within our proposed framework combine evidence from the entire network. Unlike previous OOD detection heuristics, this framework returns a \$p\$-value for each test sample. It is guaranteed to maintain the Type I Error (T1E - incorrectly predicting OOD for an actual in-distribution sample) for test data. Moreover, this allows to combine several detectors while maintaining the T1E. Building on this framework, we suggest a novel OOD procedure based on low-order statistics. Our method achieves comparable or better results than state-of-the-art methods on well-accepted OOD benchmarks, without retraining the network parameters or assuming prior knowledge on the test distribution -- and at a fraction of the computational cost.},
	urldate = {2023-12-06},
	publisher = {arXiv},
	author = {Haroush, Matan and Frostig, Tzviel and Heller, Ruth and Soudry, Daniel},
	month = mar,
	year = {2022},
	note = {arXiv:2102.12967 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{grosse_statistical_2017,
	title = {On the ({Statistical}) {Detection} of {Adversarial} {Examples}},
	url = {http://arxiv.org/abs/1702.06280},
	abstract = {Machine Learning (ML) models are applied in a variety of tasks such as network intrusion detection or Malware classification. Yet, these models are vulnerable to a class of malicious inputs known as adversarial examples. These are slightly perturbed inputs that are classified incorrectly by the ML model. The mitigation of these adversarial inputs remains an open problem. As a step towards understanding adversarial examples, we show that they are not drawn from the same distribution than the original data, and can thus be detected using statistical tests. Using thus knowledge, we introduce a complimentary approach to identify specific inputs that are adversarial. Specifically, we augment our ML model with an additional output, in which the model is trained to classify all adversarial inputs. We evaluate our approach on multiple adversarial example crafting methods (including the fast gradient sign and saliency map methods) with several datasets. The statistical test flags sample sets containing adversarial inputs confidently at sample sizes between 10 and 100 data points. Furthermore, our augmented model either detects adversarial examples as outliers with high accuracy ({\textgreater} 80\%) or increases the adversary's cost - the perturbation added - by more than 150\%. In this way, we show that statistical properties of adversarial examples are essential to their detection.},
	urldate = {2023-12-05},
	publisher = {arXiv},
	author = {Grosse, Kathrin and Manoharan, Praveen and Papernot, Nicolas and Backes, Michael and McDaniel, Patrick},
	month = oct,
	year = {2017},
	note = {arXiv:1702.06280 [cs, stat]},
	keywords = {Computer Science - Cryptography and Security, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{eastwood_probable_nodate,
	title = {Probable {Domain} {Generalization} via {Quantile} {Risk} {Minimization}},
	abstract = {Domain generalization (DG) seeks predictors which perform well on unseen test distributions by leveraging data drawn from multiple related training distributions or domains. To achieve this, DG is commonly formulated as an average- or worstcase problem over the set of possible domains. However, predictors that perform well on average lack robustness while predictors that perform well in the worst case tend to be overly-conservative. To address this, we propose a new probabilistic framework for DG where the goal is to learn predictors that perform well with high probability. Our key idea is that distribution shifts seen during training should inform us of probable shifts at test time, which we realize by explicitly relating training and test domains as draws from the same underlying meta-distribution. To achieve probable DG, we propose a new optimization problem called Quantile Risk Minimization (QRM). By minimizing the α-quantile of predictor’s risk distribution over domains, QRM seeks predictors that perform well with probability α. To solve QRM in practice, we propose the Empirical QRM (EQRM) algorithm and provide: (i) a generalization bound for EQRM; and (ii) the conditions under which EQRM recovers the causal predictor as α → 1. In our experiments, we introduce a more holistic quantile-focused evaluation protocol for DG, and demonstrate that EQRM outperforms state-of-the-art baselines on datasets from WILDS and DomainBed.},
	language = {en},
	author = {Eastwood, Cian and Robey, Alexander and Singh, Shashank},
}

@article{couso_statistical_2014,
	title = {Statistical reasoning with set-valued information: {Ontic} vs. epistemic views},
	volume = {55},
	issn = {0888613X},
	shorttitle = {Statistical reasoning with set-valued information},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X13001618},
	doi = {10.1016/j.ijar.2013.07.002},
	abstract = {In information processing tasks, sets may have a conjunctive or a disjunctive reading. In the conjunctive reading, a set represents an object of interest and its elements are subparts of the object, forming a composite description. In the disjunctive reading, a set contains mutually exclusive elements and refers to the representation of incomplete knowledge. It does not model an actual object or quantity, but partial information about an underlying object or a precise quantity. This distinction between what we call ontic vs. epistemic sets remains valid for fuzzy sets, whose membership functions, in the disjunctive reading are possibility distributions, over deterministic or random values. This paper examines the impact of this distinction in statistics. We show its importance because there is a risk of misusing basic notions and tools, such as conditioning, distance between sets, variance, regression, etc. when data are set-valued. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.},
	language = {en},
	number = {7},
	urldate = {2023-12-03},
	journal = {International Journal of Approximate Reasoning},
	author = {Couso, Inés and Dubois, Didier},
	month = oct,
	year = {2014},
	pages = {1502--1518},
}

@misc{shaker_aleatoric_2020,
	title = {Aleatoric and {Epistemic} {Uncertainty} with {Random} {Forests}},
	url = {http://arxiv.org/abs/2001.00893},
	abstract = {Due to the steadily increasing relevance of machine learning for practical applications, many of which are coming with safety requirements, the notion of uncertainty has received increasing attention in machine learning research in the last couple of years. In particular, the idea of distinguishing between two important types of uncertainty, often refereed to as aleatoric and epistemic, has recently been studied in the setting of supervised learning. In this paper, we propose to quantify these uncertainties with random forests. More specifically, we show how two general approaches for measuring the learner's aleatoric and epistemic uncertainty in a prediction can be instantiated with decision trees and random forests as learning algorithms in a classification setting. In this regard, we also compare random forests with deep neural networks, which have been used for a similar purpose.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Shaker, Mohammad Hossein and Hüllermeier, Eyke},
	month = jan,
	year = {2020},
	note = {arXiv:2001.00893 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{hullermeier_quantification_2022,
	title = {Quantification of {Credal} {Uncertainty} in {Machine} {Learning}: {A} {Critical} {Analysis} and {Empirical} {Comparison}},
	shorttitle = {Quantification of {Credal} {Uncertainty} in {Machine} {Learning}},
	url = {https://proceedings.mlr.press/v180/hullermeier22a.html},
	abstract = {The representation and quantification of uncertainty has received increasing attention in machine learning in the recent past. The formalism of credal sets provides an interesting alternative in this regard, especially as it combines the representation of epistemic (lack of knowledge) and aleatoric (statistical) uncertainty in a rather natural way. In this paper, we elaborate on uncertainty measures for credal sets from the perspective of machine learning. More specifically, we provide an overview of proposals, discuss existing measures in a critical way, and also propose a new measure that is more tailored to the machine learning setting. Based on an experimental study, we conclude that theoretically well-justified measures also lead to better performance in practice. Besides, we corroborate the difficulty of the disaggregation problem, that is, of decomposing the amount of total uncertainty into aleatoric and epistemic uncertainty in a sound manner, thereby complementing theoretical findings with empirical evidence.},
	language = {en},
	urldate = {2023-12-03},
	booktitle = {Proceedings of the {Thirty}-{Eighth} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	publisher = {PMLR},
	author = {Hüllermeier, Eyke and Destercke, Sébastien and Shaker, Mohammad Hossein},
	month = aug,
	year = {2022},
	note = {ISSN: 2640-3498},
	pages = {548--557},
}

@misc{vejnarova_composition_2017,
	title = {Composition of {Credal} {Sets} via {Polyhedral} {Geometry}},
	url = {http://arxiv.org/abs/1705.03352},
	abstract = {Recently introduced composition operator for credal sets is an analogy of such operators in probability, possibility, evidence and valuation-based systems theories. It was designed to construct multidimensional models (in the framework of credal sets) from a system of low- dimensional credal sets. In this paper we study its potential from the computational point of view utilizing methods of polyhedral geometry.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Vejnarová, Jiřina and Kratochvíl, Václav},
	month = may,
	year = {2017},
	note = {arXiv:1705.03352 [cs]},
	keywords = {68T37, Computer Science - Artificial Intelligence, I.2},
}

@misc{foll_gated_2023,
	title = {Gated {Domain} {Units} for {Multi}-source {Domain} {Generalization}},
	url = {http://arxiv.org/abs/2206.12444},
	abstract = {The phenomenon of distribution shift (DS) occurs when a dataset at test time differs from the dataset at training time, which can significantly impair the performance of a machine learning model in practical settings due to a lack of knowledge about the data's distribution at test time. To address this problem, we postulate that real-world distributions are composed of latent Invariant Elementary Distributions (I.E.D) across different domains. This assumption implies an invariant structure in the solution space that enables knowledge transfer to unseen domains. To exploit this property for domain generalization, we introduce a modular neural network layer consisting of Gated Domain Units (GDUs) that learn a representation for each latent elementary distribution. During inference, a weighted ensemble of learning machines can be created by comparing new observations with the representations of each elementary distribution. Our flexible framework also accommodates scenarios where explicit domain information is not present. Extensive experiments on image, text, and graph data show consistent performance improvement on out-of-training target domains. These findings support the practicality of the I.E.D assumption and the effectiveness of GDUs for domain generalisation.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Föll, Simon and Dubatovka, Alina and Ernst, Eugen and Chau, Siu Lun and Maritsch, Martin and Okanovic, Patrik and Thäter, Gudrun and Buhmann, Joachim M. and Wortmann, Felix and Muandet, Krikamol},
	month = may,
	year = {2023},
	note = {arXiv:2206.12444 [cs]},
	keywords = {Computer Science - Machine Learning},
}

@misc{mansour_multiple_2012,
	title = {Multiple {Source} {Adaptation} and the {Renyi} {Divergence}},
	url = {http://arxiv.org/abs/1205.2628},
	abstract = {This paper presents a novel theoretical study of the general problem of multiple source adaptation using the notion of Renyi divergence. Our results build on our previous work [12], but significantly broaden the scope of that work in several directions. We extend previous multiple source loss guarantees based on distribution weighted combinations to arbitrary target distributions P, not necessarily mixtures of the source distributions, analyze both known and unknown target distribution cases, and prove a lower bound. We further extend our bounds to deal with the case where the learner receives an approximate distribution for each source instead of the exact one, and show that similar loss guarantees can be achieved depending on the divergence between the approximate and true distributions. We also analyze the case where the labeling functions of the source domains are somewhat different. Finally, we report the results of experiments with both an artificial data set and a sentiment analysis task, showing the performance benefits of the distribution weighted combinations and the quality of our bounds based on the Renyi divergence.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Mansour, Yishay and Mohri, Mehryar and Rostamizadeh, Afshin},
	month = may,
	year = {2012},
	note = {arXiv:1205.2628 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{hoffman_algorithms_2018,
	title = {Algorithms and {Theory} for {Multiple}-{Source} {Adaptation}},
	volume = {31},
	url = {https://proceedings.neurips.cc/paper_files/paper/2018/hash/2e2079d63348233d91cad1fa9b1361e9-Abstract.html},
	abstract = {We present a number of novel contributions to the multiple-source adaptation problem. We derive new normalized solutions with strong theoretical guarantees for the cross-entropy loss and other similar losses. We also provide new guarantees that hold in the case where the conditional probabilities for the source domains are distinct. Moreover, we give new algorithms for determining the distribution-weighted combination solution for the cross-entropy loss and other losses. We report the results of a series of experiments with real-world datasets. We find that our algorithm outperforms competing approaches by producing a single robust model that performs well on any target mixture distribution. Altogether, our theory, algorithms, and empirical results provide a full solution for the multiple-source adaptation problem with very practical benefits.},
	urldate = {2023-12-03},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Hoffman, Judy and Mohri, Mehryar and Zhang, Ningshan},
	year = {2018},
}

@misc{albuquerque_adversarial_2019,
	title = {Adversarial target-invariant representation learning for domain generalization},
	url = {http://arxiv.org/abs/1911.00804},
	abstract = {Supervised learning results typically rely on assumptions of i.i.d. data. Unfortunately, those assumptions are commonly violated in practice. In this work, we tackle such problem by focusing on domain generalization: a formalization where the data generating process at test time may yield samples from never-before-seen domains (distributions). Our work relies on the following lemma: by minimizing a notion of discrepancy between all pairs from a set of given domains, we also minimize the discrepancy between any pairs of mixtures of domains. Using this result, we derive a generalization bound for our setting. We then show that low risk over unseen domains can be achieved by representing the data in a space where (i) the training distributions are indistinguishable, and (ii) relevant information for the task at hand is preserved. Minimizing the terms in our bound yields an adversarial formulation which estimates and minimizes pairwise discrepancies. We validate our proposed strategy on standard domain generalization benchmarks, outperforming a number of recently introduced methods. Notably, we tackle a real-world application where the underlying data corresponds to multi-channel electroencephalography time series from different subjects, each considered as a distinct domain.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Albuquerque, Isabela and Monteiro, João and Darvishi, Mohammad and Falk, Tiago H. and Mitliagkas, Ioannis},
	month = nov,
	year = {2019},
	note = {arXiv:1911.00804 [cs, stat]
version: 1},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{yang_generalized_2022,
	title = {Generalized {Out}-of-{Distribution} {Detection}: {A} {Survey}},
	shorttitle = {Generalized {Out}-of-{Distribution} {Detection}},
	url = {http://arxiv.org/abs/2110.11334},
	abstract = {Out-of-distribution (OOD) detection is critical to ensuring the reliability and safety of machine learning systems. For instance, in autonomous driving, we would like the driving system to issue an alert and hand over the control to humans when it detects unusual scenes or objects that it has never seen during training time and cannot make a safe decision. The term, OOD detection, first emerged in 2017 and since then has received increasing attention from the research community, leading to a plethora of methods developed, ranging from classification-based to density-based to distance-based ones. Meanwhile, several other problems, including anomaly detection (AD), novelty detection (ND), open set recognition (OSR), and outlier detection (OD), are closely related to OOD detection in terms of motivation and methodology. Despite common goals, these topics develop in isolation, and their subtle differences in definition and problem setting often confuse readers and practitioners. In this survey, we first present a unified framework called generalized OOD detection, which encompasses the five aforementioned problems, i.e., AD, ND, OSR, OOD detection, and OD. Under our framework, these five problems can be seen as special cases or sub-tasks, and are easier to distinguish. We then review each of these five areas by summarizing their recent technical developments, with a special focus on OOD detection methodologies. We conclude this survey with open challenges and potential research directions.},
	urldate = {2023-12-03},
	publisher = {arXiv},
	author = {Yang, Jingkang and Zhou, Kaiyang and Li, Yixuan and Liu, Ziwei},
	month = aug,
	year = {2022},
	note = {arXiv:2110.11334 [cs]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning},
}

@misc{sale_is_2023,
	title = {Is the {Volume} of a {Credal} {Set} a {Good} {Measure} for {Epistemic} {Uncertainty}?},
	url = {http://arxiv.org/abs/2306.09586},
	abstract = {Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as \$d\$-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful measure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.},
	urldate = {2023-12-01},
	publisher = {arXiv},
	author = {Sale, Yusuf and Caprio, Michele and Hüllermeier, Eyke},
	month = jun,
	year = {2023},
	note = {arXiv:2306.09586 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@book{boyd_convex_2004,
	address = {Cambridge, UK ; New York},
	title = {Convex optimization},
	isbn = {978-0-521-83378-3},
	language = {en},
	publisher = {Cambridge University Press},
	author = {Boyd, Stephen P. and Vandenberghe, Lieven},
	year = {2004},
	keywords = {Convex functions, Mathematical optimization},
}

@misc{dogas_two-step_2021,
	title = {Two-step method for assessing dissimilarity of random sets},
	url = {http://arxiv.org/abs/2105.05952},
	abstract = {The paper concerns a new statistical method for assessing dissimilarity of two random sets based on one realisation of each of them. The method focuses on shapes of the components of the random sets, namely on the curvature of their boundaries together with the ratios of their perimeters and areas. Theoretical background is introduced and then, the method is described, justified by a simulation study and applied to real data of two different types of tissue - mammary cancer and mastopathy.},
	urldate = {2023-11-29},
	publisher = {arXiv},
	author = {Đogaš, Vesna Gotovac and Helisová, Kateřina and Radović, Bogdan and Staněk, Jakub and Zikmundová, Markéta and Brejchová, Kateřina},
	month = may,
	year = {2021},
	note = {arXiv:2105.05952 [stat]},
	keywords = {Statistics - Methodology},
}

@inproceedings{wang_two-sample_2022,
	title = {Two-{Sample} {Test} with {Kernel} {Projected} {Wasserstein} {Distance}},
	url = {https://proceedings.mlr.press/v151/wang22f.html},
	abstract = {We develop a kernel projected Wasserstein distance for the two-sample test, an essential building block in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. This method operates by finding the nonlinear mapping in the data space which maximizes the distance between projected distributions. In contrast to existing works about projected Wasserstein distance, the proposed method circumvents the curse of dimensionality more efficiently. We present practical algorithms for computing this distance function together with the non-asymptotic uncertainty quantification of empirical estimates. Numerical examples validate our theoretical results and demonstrate good performance of the proposed method.},
	language = {en},
	urldate = {2023-11-29},
	booktitle = {Proceedings of {The} 25th {International} {Conference} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Wang, Jie and Gao, Rui and Xie, Yao},
	month = may,
	year = {2022},
	note = {ISSN: 2640-3498},
	pages = {8022--8055},
}

@article{dogas_testing_2021,
	title = {Testing {Equality} of {Distributions} of {Random} {Convex} {Compact} {Sets} via {Theory} of \${\textbackslash}mathfrak \{{N}\}\$-{Distances}},
	volume = {23},
	issn = {1387-5841, 1573-7713},
	url = {https://link.springer.com/10.1007/s11009-019-09747-z},
	doi = {10.1007/s11009-019-09747-z},
	abstract = {This paper concerns a method of testing the equality of distributions of random convex compact sets. The main theoretical result involves a construction of a metric on the space of distributions of random convex compact sets. We obtain it by using the theory of Ndistances and the redefined characteristic function of random convex compact set. We propose an approximation of the metric through its finite-dimensional counterparts. This result leads to a new statistical test for testing the equality of distributions of two random convex compact sets. Consequently, we show a heuristic approach how to determine whether two realisations of random sets that can be approximated by a union of identically distributed random convex compact sets come from the same underlying process using the constructed test. Each procedure is justified by an extensive simulation study and the heuristic method for comparing random sets using their convex compact counterparts is moreover applied to real data concerning histological images of two different types of mammary tissue.},
	language = {en},
	number = {2},
	urldate = {2023-11-27},
	journal = {Methodology and Computing in Applied Probability},
	author = {Dogaš, Vesna Gotovac and Helisová, Kateřina},
	month = jun,
	year = {2021},
	pages = {503--526},
}

@misc{gotovac_testing_2018,
	title = {Testing equality in distribution of random convex compact sets via theory of {N}-distances and its application to assessing similarity of general random sets},
	url = {http://arxiv.org/abs/1801.02090},
	abstract = {This paper concerns a method of testing equality of distribution of random convex compact sets and the way how to use the test to distinguish between two realisations of general random sets. The family of metrics on the space of distributions of random convex compact sets is constructed using the theory of N-distances and characteristic functions of random convex compact sets. Further, the approximation of the metrics through its finite dimensional counterparts is proposed, which lead to a new statistical test for testing equality in distribution of two random convex compact sets. Then, it is described how to approximate a realisation of a general random set by a union of convex compact sets, and it is shown how to determine whether two realisations of general random sets come from the same process using the constructed test. The procedure is justified by an extensive simulation study.},
	urldate = {2023-11-27},
	publisher = {arXiv},
	author = {Gotovac, Vesna and Helisová, Kateřina},
	month = jan,
	year = {2018},
	note = {arXiv:1801.02090 [math, stat]},
	keywords = {28C15, 62M30, Mathematics - Statistics Theory},
}

@book{bertsekas_convex_2009,
	address = {Belmont, Mass},
	title = {Convex optimization theory},
	isbn = {978-1-886529-31-1},
	language = {en},
	publisher = {Athena Scientific},
	editor = {Bertsekas, Dimitri P.},
	year = {2009},
}

@inproceedings{miranda_extreme_2013,
	address = {Utrecht, Netherlands},
	series = {{LNAI}},
	title = {Extreme {Points} of the {Credal} {Sets} {Generated} by {Elementary} {Comparative} {Probabilities}},
	url = {https://hal.science/hal-00861991},
	abstract = {When using convex probability sets (or, equivalently, lower previsions) as models of uncertainty, identifying extreme points can be useful to perform various computations or to use some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the elements of a finite space. We use these extreme points to study the properties of the lower probability induced by this set, and connect comparative probabilities with other uncertainty models.},
	urldate = {2023-11-27},
	booktitle = {Proceedings of {ECSQARU} 2013},
	publisher = {Springer},
	author = {Miranda, Enrique and Destercke, Sébastien},
	editor = {Gaag, Linda Van der},
	month = jul,
	year = {2013},
	keywords = {2-monotone capacities, Comparative probabilities, belief functions, credal sets, imprecise mass functions, regular extension},
	pages = {424--435},
}

@article{maua_thirty_2020,
	title = {Thirty years of credal networks: {Specification}, algorithms and complexity},
	volume = {126},
	issn = {0888613X},
	shorttitle = {Thirty years of credal networks},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X20302152},
	doi = {10.1016/j.ijar.2020.08.009},
	language = {en},
	urldate = {2023-11-27},
	journal = {International Journal of Approximate Reasoning},
	author = {Mauá, Denis Deratani and Cozman, Fabio Gagliardi},
	month = nov,
	year = {2020},
	pages = {133--157},
}

@article{cozman_evenly_2018,
	title = {Evenly convex credal sets},
	volume = {103},
	issn = {0888613X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X17305923},
	doi = {10.1016/j.ijar.2018.09.003},
	language = {en},
	urldate = {2023-11-26},
	journal = {International Journal of Approximate Reasoning},
	author = {Cozman, Fabio Gagliardi},
	month = dec,
	year = {2018},
	pages = {124--138},
}

@article{bronevich_modelling_2018,
	title = {Modelling uncertainty with generalized credal sets: application to conjunction and decision},
	volume = {47},
	issn = {0308-1079, 1563-5104},
	shorttitle = {Modelling uncertainty with generalized credal sets},
	url = {https://www.tandfonline.com/doi/full/10.1080/03081079.2017.1391805},
	doi = {10.1080/03081079.2017.1391805},
	language = {en},
	number = {1},
	urldate = {2023-11-26},
	journal = {International Journal of General Systems},
	author = {Bronevich, Andrey G. and Rozenberg, Igor N.},
	month = jan,
	year = {2018},
	pages = {67--96},
}

@misc{noauthor_modelling_nodate,
	title = {Modelling uncertainty with generalized credal sets: application to conjunction and decision},
	shorttitle = {Modelling uncertainty with generalized credal sets},
	url = {https://www.tandfonline.com/doi/epdf/10.1080/03081079.2017.1391805?needAccess=true},
	language = {en},
	urldate = {2023-11-26},
	note = {ISSN: 0308-1079},
}

@article{bronevich_contradiction_2019,
	title = {The contradiction between belief functions: {Its} description, measurement, and correction based on generalized credal sets},
	volume = {112},
	issn = {0888613X},
	shorttitle = {The contradiction between belief functions},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X18306194},
	doi = {10.1016/j.ijar.2019.06.001},
	language = {en},
	urldate = {2023-11-26},
	journal = {International Journal of Approximate Reasoning},
	author = {Bronevich, Andrey G. and Rozenberg, Igor N.},
	month = sep,
	year = {2019},
	pages = {119--139},
}

@inproceedings{vellanki_process-constrained_2017,
	title = {Process-constrained batch {Bayesian} optimisation},
	volume = {30},
	url = {https://papers.nips.cc/paper_files/paper/2017/hash/1f71e393b3809197ed66df836fe833e5-Abstract.html},
	abstract = {Abstract Prevailing batch Bayesian optimisation methods allow all control variables to be freely altered at each iteration. Real-world experiments, however, often have physical limitations making it time-consuming to alter all settings for each recommendation in a batch. This gives rise to a unique problem in BO: in a recommended batch, a set of variables that are expensive to experimentally change need to be fixed, while the remaining control variables can be varied. We formulate this as a process-constrained batch Bayesian optimisation problem. We propose two algorithms, pc-BO(basic) and pc-BO(nested). pc-BO(basic) is simpler but lacks convergence guarantee. In contrast pc-BO(nested) is slightly more complex, but admits convergence analysis. We show that the regret of pc-BO(nested) is sublinear. We demonstrate the performance of both pc-BO(basic) and pc-BO(nested) by optimising benchmark test functions, tuning hyper-parameters of the SVM classifier, optimising the heat-treatment process for an Al-Sc alloy to achieve target hardness, and optimising the short polymer fibre production process.},
	urldate = {2023-11-24},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Vellanki, Pratibha and Rana, Santu and Gupta, Sunil and Rubin, David and Sutti, Alessandra and Dorin, Thomas and Height, Murray and Sanders, Paul and Venkatesh, Svetha},
	year = {2017},
}

@misc{daulton_differentiable_2020,
	title = {Differentiable {Expected} {Hypervolume} {Improvement} for {Parallel} {Multi}-{Objective} {Bayesian} {Optimization}},
	url = {http://arxiv.org/abs/2006.05078},
	abstract = {In many real-world scenarios, decision makers seek to efficiently optimize multiple competing objectives in a sample-efficient fashion. Multi-objective Bayesian optimization (BO) is a common approach, but many of the best-performing acquisition functions do not have known analytic gradients and suffer from high computational overhead. We leverage recent advances in programming models and hardware acceleration for multi-objective BO using Expected Hypervolume Improvement (EHVI)---an algorithm notorious for its high computational complexity. We derive a novel formulation of q-Expected Hypervolume Improvement (qEHVI), an acquisition function that extends EHVI to the parallel, constrained evaluation setting. qEHVI is an exact computation of the joint EHVI of q new candidate points (up to Monte-Carlo (MC) integration error). Whereas previous EHVI formulations rely on gradient-free acquisition optimization or approximated gradients, we compute exact gradients of the MC estimator via auto-differentiation, thereby enabling efficient and effective optimization using first-order and quasi-second-order methods. Our empirical evaluation demonstrates that qEHVI is computationally tractable in many practical scenarios and outperforms state-of-the-art multi-objective BO algorithms at a fraction of their wall time.},
	urldate = {2023-11-24},
	publisher = {arXiv},
	author = {Daulton, Samuel and Balandat, Maximilian and Bakshy, Eytan},
	month = oct,
	year = {2020},
	note = {arXiv:2006.05078 [cs, math, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Mathematics - Optimization and Control, Statistics - Machine Learning},
}

@article{wills_hausdor_nodate,
	title = {Hausdorﬀ {Distance} and {Convex} {Sets}},
	language = {en},
	author = {Wills, M D},
}

@article{alparslan_gok_interval_2010,
	title = {The interval {Shapley} value: an axiomatization},
	volume = {18},
	issn = {1435-246X, 1613-9178},
	shorttitle = {The interval {Shapley} value},
	url = {http://link.springer.com/10.1007/s10100-009-0096-0},
	doi = {10.1007/s10100-009-0096-0},
	abstract = {The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was deﬁned and axiomatically characterized in different game-theoretic models. Recently much research work has been done in order to extend OR models and methods, in particular cooperative game theory, for situations with interval data. This paper focuses on the Shapley value for cooperative games where the set of players is ﬁnite and the coalition values are compact intervals of real numbers. The interval Shapley value is characterized with the aid of the properties of additivity, efﬁciency, symmetry and dummy player, which are straightforward generalizations of the corresponding properties in the classical cooperative game theory.},
	language = {en},
	number = {2},
	urldate = {2023-11-23},
	journal = {Central European Journal of Operations Research},
	author = {Alparslan Gök, S. Z. and Branzei, R. and Tijs, S.},
	month = jun,
	year = {2010},
	pages = {131--140},
}

@article{bronevich_modelling_2018-1,
	title = {Modelling uncertainty with generalized credal sets: application to conjunction and decision},
	volume = {47},
	issn = {0308-1079, 1563-5104},
	shorttitle = {Modelling uncertainty with generalized credal sets},
	url = {https://www.tandfonline.com/doi/full/10.1080/03081079.2017.1391805},
	doi = {10.1080/03081079.2017.1391805},
	language = {en},
	number = {1},
	urldate = {2023-11-19},
	journal = {International Journal of General Systems},
	author = {Bronevich, Andrey G. and Rozenberg, Igor N.},
	month = jan,
	year = {2018},
	pages = {67--96},
}

@article{dosovitskiy_you_2020,
	title = {{YOU} {ONLY} {TRAIN} {ONCE}: {LOSS}-{CONDITIONAL} {TRAINING} {OF} {DEEP} {NETWORKS}},
	abstract = {In many machine learning problems, loss functions are weighted sums of several terms. A typical approach to dealing with these is to train multiple separate models with different selections of weights and then either choose the best one according to some criterion or keep multiple models if it is desirable to maintain a diverse set of solutions. This is inefﬁcient both at training and at inference time. We propose a method that allows replacing multiple models trained on one loss function each by a single model trained on a distribution of losses. At test time a model trained this way can be conditioned to generate outputs corresponding to any loss from the training distribution of losses. We demonstrate this approach on three tasks with parameterized losses: β-VAE, learned image compression, and fast style transfer.},
	language = {en},
	author = {Dosovitskiy, Alexey and Djolonga, Josip},
	year = {2020},
}

@book{williamson_defence_2010,
	title = {In {Defence} of {Objective} {Bayesianism}},
	isbn = {978-0-19-922800-3},
	url = {http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199228003.001.0001/acprof-9780199228003},
	abstract = {Section. 8.1 introduces the problem of judgement aggregation and some of the di culties encountered in trying to solve this problem. Section. 8.2 introduces the theory of belief revision and merging and Pigozzi's application of this theory to judgement aggregation. Section. 8.3 argues that judgements should not be merged directly; rather, one should merge the evidence on which the judgements are based. Given this merged evidence, objective Bayesianism and decision theory can be used to derive an appropriate set of judgements—the resulting judgements should be viewed as the aggregate of the judgements of the original individuals (§8.4).},
	language = {en},
	urldate = {2023-11-15},
	publisher = {Oxford University Press},
	author = {Williamson, Jon},
	month = apr,
	year = {2010},
	doi = {10.1093/acprof:oso/9780199228003.001.0001},
	doi = {10.1093/acprof:oso/9780199228003.001.0001},
}

@article{williamson_2_nodate,
	title = {2 2 {Objective} {Bayesianism} },
	abstract = {This chapter reviews the various interpretations of probability (§2.2), assessing them in the light of certain desiderata for a philosophical theory of probability that are presented in §2.1. It is suggested that present‐day objective Bayesianism, which is characterized in §2.3, is close to early views of probability, such as those of Jakob Bernoulli and Thomas Bayes.},
	language = {en},
	author = {Williamson, Jon},
}

@article{lin_robustness_nodate,
	title = {On the {Robustness} of {Removal}-{Based} {Feature} {Attributions}},
	abstract = {To explain predictions made by complex machine learning models, many feature attribution methods have been developed that assign importance scores to input features. Some recent work challenges the robustness of these methods by showing that they are sensitive to input and model perturbations, while other work addresses this issue by proposing robust attribution methods. However, previous work on attribution robustness has focused primarily on gradient-based feature attributions, whereas the robustness of removal-based attribution methods is not currently well understood. To bridge this gap, we theoretically characterize the robustness properties of removal-based feature attributions. Specifically, we provide a unified analysis of such methods and derive upper bounds for the difference between intact and perturbed attributions, under settings of both input and model perturbations. Our empirical results on synthetic and real-world data validate our theoretical results and demonstrate their practical implications, including the ability to increase attribution robustness by improving the model’s Lipschitz regularity.},
	language = {en},
	author = {Lin, Chris and Covert, Ian},
}

@article{mork_uncertainty_2013,
	title = {Uncertainty, credal sets and second order probability},
	volume = {190},
	issn = {0039-7857, 1573-0964},
	url = {http://link.springer.com/10.1007/s11229-011-0042-2},
	doi = {10.1007/s11229-011-0042-2},
	abstract = {The last 20 years or so has seen an intense search carried out within Dempster–Shafer theory, with the aim of ﬁnding a generalization of the Shannon entropy for belief functions. In that time, there has also been much progress made in credal set theory—another generalization of the traditional Bayesian epistemic representation—albeit not in this particular area. In credal set theory, sets of probability functions are utilized to represent the epistemic state of rational agents instead of the single probability function of traditional Bayesian theory. The Shannon entropy has been shown to uniquely capture certain highly intuitive properties of uncertainty, and can thus be considered a measure of that quantity. This article presents two measures developed with the purpose of generalizing the Shannon entropy for (1) unordered convex credal sets and (2) possibly non-convex credal sets ordered by second order probability, thereby providing uncertainty measures for such epistemic representations. There is also a comparison with the results of the measure AU developed within Dempster–Shafer theory in a few instances where unordered convex credal set theory and Dempster–Shafer theory overlap.},
	language = {en},
	number = {3},
	urldate = {2023-11-15},
	journal = {Synthese},
	author = {Mork, Jonas Clausen},
	month = feb,
	year = {2013},
	pages = {353--378},
}

@article{cozman_sets_2012,
	title = {Sets of probability distributions, independence, and convexity},
	volume = {186},
	issn = {0039-7857, 1573-0964},
	url = {http://link.springer.com/10.1007/s11229-011-9999-0},
	doi = {10.1007/s11229-011-9999-0},
	abstract = {This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions. The starting point is Kyburg and Pittarelli’s discussion of “convex Bayesianism” (in particular their proposals concerning E-admissibility, independence, and convexity). The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justiﬁcation of “strong independence” (using exchangeability) are presented. Finally, the connection between Kyburg and Pittarelli’s results and recent developments on the axiomatization of non-binary preferences, and its impact on “complete” independence, are described.},
	language = {en},
	number = {2},
	urldate = {2023-11-15},
	journal = {Synthese},
	author = {Cozman, Fabio G.},
	month = may,
	year = {2012},
	pages = {577--600},
}

@article{howson_epistemic_1996,
	title = {Epistemic {Probability}},
	volume = {70},
	issn = {0309-7013},
	url = {https://www.jstor.org/stable/4107003},
	urldate = {2023-11-14},
	journal = {Proceedings of the Aristotelian Society, Supplementary Volumes},
	author = {Howson, Colin and Castell, Paul},
	year = {1996},
	note = {Publisher: [Aristotelian Society, Wiley]},
	pages = {63--94},
}

@article{van_fraassen_rational_1980,
	title = {Rational {Belief} and {Probability} {Kinematics}},
	volume = {47},
	issn = {0031-8248},
	url = {https://www.jstor.org/stable/187082},
	abstract = {A general form is proposed for epistemological theories, the relevant factors being: the family of epistemic judgments, the epistemic state, the epistemic commitment (governing change of state), and the family of possible epistemic inputs (deliverances of experience). First a simple theory is examined in which the states are probability functions, and the subject of probability kinematics introduced by Richard Jeffrey is explored. Then a second theory is examined in which the state has as constituents a body of information (rational corpus) and a recipe that determines the accepted epistemic judgments on the basis of this corpus. Through an examination of several approaches to the statistical syllogism, a relation is again established with Jeffrey's generalized conditionalization.},
	number = {2},
	urldate = {2023-11-14},
	journal = {Philosophy of Science},
	author = {Van Fraassen, Bas C.},
	year = {1980},
	note = {Publisher: [The University of Chicago Press, Philosophy of Science Association]},
	pages = {165--187},
}

@article{akbari_causal_nodate,
	title = {Causal {Effect} {Identification} in {Uncertain} {Causal} {Networks}},
	abstract = {Causal identification is at the core of the causal inference literature, where complete algorithms have been proposed to identify causal queries of interest. The validity of these algorithms hinges on the restrictive assumption of having access to a correctly specified causal structure. In this work, we study the setting where a probabilistic model of the causal structure is available. Specifically, the edges in a causal graph exist with uncertainties which may, for example, represent degree of belief from domain experts. Alternatively, the uncertainty about an edge may reflect the confidence of a particular statistical test. The question that naturally arises in this setting is: Given such a probabilistic graph and a specific causal effect of interest, what is the subgraph which has the highest plausibility and for which the causal effect is identifiable? We show that answering this question reduces to solving an NP-complete combinatorial optimization problem which we call the edge ID problem. We propose efficient algorithms to approximate this problem and evaluate them against both real-world networks and randomly generated graphs.},
	language = {en},
	author = {Akbari, Sina and Jamshidi, Fateme and Mokhtarian, Ehsan},
}

@misc{wang_two-sample_2022-1,
	title = {Two-sample {Test} with {Kernel} {Projected} {Wasserstein} {Distance}},
	url = {http://arxiv.org/abs/2102.06449},
	abstract = {We develop a kernel projected Wasserstein distance for the two-sample test, an essential building block in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. This method operates by finding the nonlinear mapping in the data space which maximizes the distance between projected distributions. In contrast to existing works about projected Wasserstein distance, the proposed method circumvents the curse of dimensionality more efficiently. We present practical algorithms for computing this distance function together with the non-asymptotic uncertainty quantification of empirical estimates. Numerical examples validate our theoretical results and demonstrate good performance of the proposed method.},
	urldate = {2023-11-14},
	publisher = {arXiv},
	author = {Wang, Jie and Gao, Rui and Xie, Yao},
	month = feb,
	year = {2022},
	note = {arXiv:2102.06449 [math, stat]},
	keywords = {Mathematics - Statistics Theory},
}

@article{bergemann_information_2019,
	title = {Information {Design}: {A} {Unified} {Perspective}},
	volume = {57},
	issn = {0022-0515},
	shorttitle = {Information {Design}},
	url = {https://pubs.aeaweb.org/doi/10.1257/jel.20181489},
	doi = {10.1257/jel.20181489},
	abstract = {Given a game with uncertain payoffs, information design analyzes the extent to which the provision of information alone can influence the behavior of the players. Information design has a literal interpretation, under which there is a real information designer who can commit to the choice of the best information structure (from her perspective) for a set of participants in a game. We emphasize a metaphorical interpretation, under which the information design problem is used by the analyst to characterize play in the game under many different information structures. We provide an introduction to the basic issues and insights of a rapidly growing literature in information design. We show how the literal and metaphorical interpretations of information design unify a large body of existing work, including that on communication in games (Myerson 1991), Bayesian persuasion (Kamenica and Gentzkow 2011), and some of our own recent work on robust predictions in games of incomplete information. ( JEL C70, D82, D83)},
	language = {en},
	number = {1},
	urldate = {2023-11-13},
	journal = {Journal of Economic Literature},
	author = {Bergemann, Dirk and Morris, Stephen},
	month = mar,
	year = {2019},
	pages = {44--95},
}

@article{giron_quasi-bayesian_1980,
	title = {Quasi-{Bayesian} {Behaviour}: {A} more realistic approach to decision making?},
	volume = {31},
	issn = {0041-0241},
	shorttitle = {Quasi-{Bayesian} {Behaviour}},
	url = {http://link.springer.com/10.1007/BF02888345},
	doi = {10.1007/BF02888345},
	abstract = {In this paper the theoretical and practical implications of dropping-from the basic Bayesian coherence principles- the assumption of comparability of every pair of acts is examined. The resulting theory is shown to be still perfectly coherent and has Bayesian theory as a particular case. In particular we question the need of weakening or ruling out some of the axioms that constitute the coherence principles; what are their practical implications; how this drive to the notion of partial information or partial uncertainty in a certain sense; how this partial information is combined with sample information and how this relates to Bayesian methods. We also point out the relation of this approach to rational behaviour with the more (and apparently unrelated) general notion of domination structures as applied to multicriteria decision making.},
	language = {en},
	number = {1},
	urldate = {2023-11-13},
	journal = {Trabajos de Estadistica Y de Investigacion Operativa},
	author = {Giron, F. J. and Rios, S.},
	month = feb,
	year = {1980},
	pages = {17--38},
}

@incollection{wiedermann_evidence_2016,
	edition = {1},
	title = {Evidence and {Epistemic} {Causality}},
	isbn = {978-1-118-94704-3 978-1-118-94707-4},
	url = {https://onlinelibrary.wiley.com/doi/10.1002/9781118947074.ch2},
	language = {en},
	urldate = {2023-11-10},
	booktitle = {Wiley {Series} in {Probability} and {Statistics}},
	publisher = {Wiley},
	author = {Wilde, Michael and Williamson, Jon},
	editor = {Wiedermann, Wolfgang and Eye, Alexander},
	month = jun,
	year = {2016},
	doi = {10.1002/9781118947074.ch2},
	pages = {31--41},
}

@article{williamson_calibration_2021,
	title = {Calibration for {Epistemic} {Causality}},
	volume = {86},
	issn = {0165-0106, 1572-8420},
	url = {https://link.springer.com/10.1007/s10670-019-00139-w},
	doi = {10.1007/s10670-019-00139-w},
	abstract = {The epistemic theory of causality is analogous to epistemic theories of probability. Most proponents of epistemic probability would argue that one’s degrees of belief should be calibrated to chances, insofar as one has evidence of chances. The question arises as to whether causal beliefs should satisfy an analogous calibration norm. In this paper, I formulate a particular version of a norm requiring calibration to chances and argue that this norm is the most fundamental evidential norm for epistemic probability. I then develop an analogous calibration norm for epistemic causality, argue that it is the only evidential norm required for epistemic causality, and show how an epistemic account of causality that incorporates this norm can be used to analyse objective causal relationships.},
	language = {en},
	number = {4},
	urldate = {2023-11-10},
	journal = {Erkenntnis},
	author = {Williamson, Jon},
	month = aug,
	year = {2021},
	pages = {941--960},
}

@article{mebane_causal_2013,
	title = {Causal {Inference} without {Ignorability}: {Identification} with {Nonrandom} {Assignment} and {Missing} {Treatment} {Data}},
	volume = {21},
	issn = {1047-1987},
	shorttitle = {Causal {Inference} without {Ignorability}},
	url = {https://www.jstor.org/stable/24572656},
	abstract = {How a treatment causes a particular outcome is a focus of inquiry in political science. When treatment data are either nonrandomly assigned or missing, the analyst will often invoke ignorability assumptions: that is, both the treatment and missingness are assumed to be as if randomly assigned, perhaps conditional on a set of observed covariates. But what if these assumptions are wrong? What if the analyst does not know why–or even if–a particular subject received a treatment? Building on Manski, Molinari offers an approach for calculating nonparametric identification bounds for the average treatment effect of a binary treatment under general missingness or nonrandom assignment. To make these bounds substantively more informative, Molinari's technique permits adding monotonicity assumptions (e.g., assuming that treatment effects are weakly positive). Given the potential importance of these assumptions, we develop a new Bayesian method for performing sensitivity analysis regarding them. This sensitivity analysis allows analysts to interpret the assumptions' consequences quantitatively and visually. We apply this method to two problems in political science, highlighting the method's utility for applied research.},
	number = {2},
	urldate = {2023-11-10},
	journal = {Political Analysis},
	author = {Mebane, Walter R. and Poast, Paul},
	year = {2013},
	note = {Publisher: [Oxford University Press, Society for Political Methodology]},
	pages = {233--251},
}

@article{spiegler_behavioral_nodate,
	title = {Behavioral {Causal} {Inference}},
	abstract = {When inferring the causal eﬀect of one variable on another from correlational data, a common practice by professional researchers as well as lay decision makers is to control for some set of exogenous confounding variables. Choosing an inappropriate set of control variables can lead to erroneous causal inferences. This paper presents a model of lay decision makers who use long-run observational data to learn the causal eﬀect of their actions on a payoﬀ-relevant outcome. Diﬀerent types of decision makers use diﬀerent sets of control variables. I obtain upper bounds on the equilibrium welfare loss due to wrong causal inferences, for various families of data-generating processes. The bounds depend on the structure of the type space. When types are “ordered” in a certain sense, the equilibrium condition greatly reduces the cost of wrong causal inference due to poor controls.},
	language = {en},
	author = {Spiegler, Ran},
}

@article{minsker_geometric_2015,
	title = {Geometric median and robust estimation in {Banach} spaces},
	volume = {21},
	issn = {1350-7265},
	url = {http://arxiv.org/abs/1308.1334},
	doi = {10.3150/14-BEJ645},
	abstract = {In many real-world applications, collected data are contaminated by noise with heavy-tailed distribution and might contain outliers of large magnitude. In this situation, it is necessary to apply methods which produce reliable outcomes even if the input contains corrupted measurements. We describe a general method which allows one to obtain estimators with tight concentration around the true parameter of interest taking values in a Banach space. Suggested construction relies on the fact that the geometric median of a collection of independent "weakly concentrated" estimators satisfies a much stronger deviation bound than each individual element in the collection. Our approach is illustrated through several examples, including sparse linear regression and low-rank matrix recovery problems.},
	number = {4},
	urldate = {2023-10-30},
	journal = {Bernoulli},
	author = {Minsker, Stanislav},
	month = nov,
	year = {2015},
	note = {arXiv:1308.1334 [math, stat]},
	keywords = {Mathematics - Statistics Theory},
}

@inproceedings{kolouri_generalized_2022,
	title = {Generalized {Sliced} {Probability} {Metrics}},
	url = {https://ieeexplore.ieee.org/document/9746016},
	doi = {10.1109/ICASSP43922.2022.9746016},
	abstract = {Sliced probability metrics have become increasingly popular in machine learning, and they play a quintessential role in various applications, including statistical hypothesis testing and generative modeling. However, in a practical setting, the convergence behavior of the algorithms built upon these distances have not been well established, except for a few specific cases. In this paper, we introduce a new family of sliced probability metrics, namely Generalized Sliced Probability Metrics (GSPMs), based on the idea of slicing high-dimensional distributions into a set of their one-dimensional marginals. We show that GSPMs are true metrics, and they are related to the Maximum Mean Discrepancy (MMD). Exploiting this relationship, we consider GSPM-based gradient flows and show that, under mild assumptions, the gradient flow converges to the global optimum. Finally, we demonstrate that various choices of GSPMs lead to new positive definite kernels that could be used in the MMD formulation while providing a unique integral geometric interpretation. We illustrate the application of GSPMs in gradient flows.},
	urldate = {2023-10-24},
	booktitle = {{ICASSP} 2022 - 2022 {IEEE} {International} {Conference} on {Acoustics}, {Speech} and {Signal} {Processing} ({ICASSP})},
	author = {Kolouri, Soheil and Nadjahi, Kimia and Shahrampour, Shahin and Şimşekli, Umut},
	month = may,
	year = {2022},
	note = {ISSN: 2379-190X},
	pages = {4513--4517},
}

@article{mansourvar_-divergence_2022,
	title = {The ϕ-divergence family of measures based on quantile function},
	volume = {56},
	issn = {0233-1888},
	url = {https://doi.org/10.1080/02331888.2022.2105844},
	doi = {10.1080/02331888.2022.2105844},
	abstract = {The quantile function can be considered an efficient alternative to the distribution function in cases where the quantile functions are tractable while the distribution functions do not have explicit forms. In this paper, we introduce a quantile-based definition of the ϕ-divergence family of measures to assess the discrepancy between two random variables. The proposed measure defines a rich family of divergence measures and includes various common divergences as special cases. Some properties of these measures of divergence are studied and several examples are also provided. Dynamic versions of the quantile-based ϕ-divergence family of measures between residual and past lifetimes are developed and their properties are studied. The suggested divergence measures are also examined in a general class of transformed models which results in some well-known models in the lifetime studies and survival analysis. Furthermore, the non-parametric estimations of the proposed divergence measures are discussed, and the performance of the resulting estimators is evaluated via simulation.},
	number = {5},
	urldate = {2023-10-24},
	journal = {Statistics},
	author = {Mansourvar, Zahra},
	month = sep,
	year = {2022},
	note = {Publisher: Taylor \& Francis
\_eprint: https://doi.org/10.1080/02331888.2022.2105844},
	keywords = {Csiszár's family of measures, Divergence measure, quantile function, transformation models},
	pages = {1113--1132},
}

@inproceedings{zhang_quantile_2020,
	title = {Quantile {Propagation} for {Wasserstein}-{Approximate} {Gaussian} {Processes}},
	volume = {33},
	url = {https://proceedings.neurips.cc/paper_files/paper/2020/hash/f5e62af885293cf4d511ceef31e61c80-Abstract.html},
	urldate = {2023-10-24},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Zhang, Rui and Walder, Christian and Bonilla, Edwin V and Rizoiu, Marian-Andrei and Xie, Lexing},
	year = {2020},
	pages = {21566--21578},
}

@misc{nadjahi_statistical_2022,
	title = {Statistical and {Topological} {Properties} of {Sliced} {Probability} {Divergences}},
	url = {http://arxiv.org/abs/2003.05783},
	abstract = {The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between one-dimensional random projections of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.},
	urldate = {2023-10-24},
	publisher = {arXiv},
	author = {Nadjahi, Kimia and Durmus, Alain and Chizat, Lénaïc and Kolouri, Soheil and Shahrampour, Shahin and Şimşekli, Umut},
	month = jan,
	year = {2022},
	note = {arXiv:2003.05783 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{noauthor_-divergence_nodate,
	title = {The ϕ-divergence family of measures based on quantile function},
	url = {https://www.tandfonline.com/doi/epdf/10.1080/02331888.2022.2105844?needAccess=true},
	language = {en},
	urldate = {2023-10-24},
	note = {ISSN: 0233-1888},
}

@misc{nadjahi_statistical_2022-1,
	title = {Statistical and {Topological} {Properties} of {Sliced} {Probability} {Divergences}},
	url = {http://arxiv.org/abs/2003.05783},
	doi = {10.48550/arXiv.2003.05783},
	abstract = {The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base divergence' between one-dimensional random projections of the two measures. However, the topological, statistical, and computational consequences of this technique have not yet been well-established. In this paper, we aim at bridging this gap and derive various theoretical properties of sliced probability divergences. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of a sliced divergence does not depend on the problem dimension. We finally apply our general results to several base divergences, and illustrate our theory on both synthetic and real data experiments.},
	urldate = {2023-10-24},
	publisher = {arXiv},
	author = {Nadjahi, Kimia and Durmus, Alain and Chizat, Lénaïc and Kolouri, Soheil and Shahrampour, Shahin and Şimşekli, Umut},
	month = jan,
	year = {2022},
	note = {arXiv:2003.05783 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{barp_targeted_2022,
	title = {Targeted {Separation} and {Convergence} with {Kernel} {Discrepancies}},
	url = {http://arxiv.org/abs/2209.12835},
	abstract = {Maximum mean discrepancies (MMDs) like the kernel Stein discrepancy (KSD) have grown central to a wide range of applications, including hypothesis testing, sampler selection, distribution approximation, and variational inference. In each setting, these kernel-based discrepancy measures are required to (i) separate a target P from other probability measures or even (ii) control weak convergence to P. In this article we derive new sufficient and necessary conditions to ensure (i) and (ii). For MMDs on separable metric spaces, we characterize those kernels that separate Bochner embeddable measures and introduce simple conditions for separating all measures with unbounded kernels and for controlling convergence with bounded kernels. We use these results on \${\textbackslash}mathbb\{R\}{\textasciicircum}d\$ to substantially broaden the known conditions for KSD separation and convergence control and to develop the first KSDs known to exactly metrize weak convergence to P. Along the way, we highlight the implications of our results for hypothesis testing, measuring and improving sample quality, and sampling with Stein variational gradient descent.},
	urldate = {2023-10-04},
	publisher = {arXiv},
	author = {Barp, Alessandro and Simon-Gabriel, Carl-Johann and Girolami, Mark and Mackey, Lester},
	month = sep,
	year = {2022},
	note = {arXiv:2209.12835 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Statistics Theory, Statistics - Machine Learning},
}

@inproceedings{christmann_how_2007,
	title = {How {SVMs} can estimate quantiles and the median},
	volume = {20},
	url = {https://proceedings.neurips.cc/paper_files/paper/2007/hash/1be3bc32e6564055d5ca3e5a354acbef-Abstract.html},
	abstract = {We investigate quantile regression based on the pinball loss and the ǫ-insensitive loss. For the pinball loss a condition on the data-generating distribution P is given that ensures that the conditional quantiles are approximated with respect to k · k1. This result is then used to derive an oracle inequality for an SVM based on the pinball loss. Moreover, we show that SVMs based on the ǫ-insensitive loss estimate the conditional median only under certain conditions on P .},
	urldate = {2023-09-26},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Christmann, Andreas and Steinwart, Ingo},
	year = {2007},
}

@inproceedings{thomas_calibration_2015,
	address = {Campus des Cordeliers, Paris, France},
	title = {Calibration of {One}-{Class} {SVM} for {MV} set estimation},
	isbn = {978-1-4673-8272-4},
	url = {http://ieeexplore.ieee.org/document/7344789/},
	doi = {10.1109/DSAA.2015.7344789},
	abstract = {A general approach for anomaly detection or novelty detection consists in estimating high density regions or Minimum Volume (MV) sets. The One-Class Support Vector Machine (OCSVM) is a state-of-the-art algorithm for estimating such regions from high dimensional data. Yet it suffers from practical limitations. When applied to a limited number of samples it can lead to poor performance even when picking the best hyperparameters. Moreover the solution of OCSVM is very sensitive to the selection of hyperparameters which makes it hard to optimize in an unsupervised setting. We present a new approach to estimate MV sets using the OCSVM with a different choice of the parameter controlling the proportion of outliers. The solution function of the OCSVM is learnt on a training set and the desired probability mass is obtained by adjusting the offset on a test set to prevent overﬁtting. Models learnt on different train/test splits are then aggregated to reduce the variance induced by such random splits. Our approach makes it possible to tune the hyperparameters automatically and obtain nested set estimates. Experimental results show that our approach outperforms the standard OCSVM formulation while suffering less from the curse of dimensionality than kernel density estimates. Results on actual data sets are also presented.},
	language = {en},
	urldate = {2023-09-26},
	booktitle = {2015 {IEEE} {International} {Conference} on {Data} {Science} and {Advanced} {Analytics} ({DSAA})},
	publisher = {IEEE},
	author = {Thomas, Albert and Feuillard, Vincent and Gramfort, Alexandre},
	month = oct,
	year = {2015},
	pages = {1--9},
}

@inproceedings{scholkopf_support_1999,
	title = {Support {Vector} {Method} for {Novelty} {Detection}},
	volume = {12},
	url = {https://proceedings.neurips.cc/paper_files/paper/1999/hash/8725fb777f25776ffa9076e44fcfd776-Abstract.html},
	abstract = {Suppose you are given some dataset drawn from an underlying probabil(cid:173) ity distribution P  and you want to estimate a "simple" subset S  of input  space such that the probability that a test point drawn from P lies outside  of S equals some a priori specified l/ between 0 and 1.  We  propose a method to  approach this  problem by trying to estimate a  function f  which is positive on S  and negative on the complement.  The  functional form of f  is given by a kernel expansion in terms of a poten(cid:173) tially small subset of the training data; it is regularized by controlling the  length of the weight vector in  an associated feature space.  We provide a  theoretical analysis of the statistical performance of our algorithm.  The algorithm is  a natural extension of the  support vector algorithm to  the case of unlabelled data.},
	urldate = {2023-09-26},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {MIT Press},
	author = {Schölkopf, Bernhard and Williamson, Robert C and Smola, Alex and Shawe-Taylor, John and Platt, John},
	year = {1999},
}

@misc{lin_causal_2022,
	title = {Causal {Inference} on {Distribution} {Functions}},
	url = {http://arxiv.org/abs/2101.01599},
	abstract = {Understanding causal relationships is one of the most important goals of modern science. So far, the causal inference literature has focused almost exclusively on outcomes coming from the Euclidean space \${\textbackslash}mathbb\{R\}{\textasciicircum}p\$. However, it is increasingly common that complex datasets are best summarized as data points in non-linear spaces. In this paper, we present a novel framework of causal effects for outcomes from the Wasserstein space of cumulative distribution functions, which in contrast to the Euclidean space, is non-linear. We develop doubly robust estimators and associated asymptotic theory for these causal effects. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.},
	urldate = {2023-09-26},
	publisher = {arXiv},
	author = {Lin, Zhenhua and Kong, Dehan and Wang, Linbo},
	month = oct,
	year = {2022},
	note = {arXiv:2101.01599 [stat]},
	keywords = {Statistics - Methodology},
}

@article{fraiman_quantiles_2012,
	title = {Quantiles for finite and infinite dimensional data},
	volume = {108},
	issn = {0047-259X},
	url = {https://www.sciencedirect.com/science/article/pii/S0047259X12000176},
	doi = {10.1016/j.jmva.2012.01.016},
	abstract = {A new projection-based definition of quantiles in a multivariate setting is proposed. This approach extends in a natural way to infinite-dimensional Hilbert spaces. The directional quantiles we define are shown to satisfy desirable properties of equivariance and, from an interpretation point of view, the resulting quantile contours provide valuable information when plotting them. Sample quantiles estimating the corresponding population quantiles are defined and consistency results are obtained. The new concept of principal quantile directions, closely related in some situations to principal component analysis, is found specially attractive for reducing the dimensionality and visualizing important features of functional data. Asymptotic properties of the empirical version of principal quantile directions are also obtained. Based on these ideas, a simple definition of robust principal components for finite and infinite-dimensional spaces is also proposed. The presented methodology is illustrated with examples throughout the paper.},
	urldate = {2023-09-26},
	journal = {Journal of Multivariate Analysis},
	author = {Fraiman, Ricardo and Pateiro-López, Beatriz},
	month = jul,
	year = {2012},
	keywords = {High dimensional multivariate data, Hilbert space, Principal quantile directions, Quantiles for functional data},
	pages = {1--14},
}

@article{vert_consistency_nodate,
	title = {Consistency and {Convergence} {Rates} of {One}-{Class} {SVMs} and {Related} {Algorithms}},
	abstract = {We determine the asymptotic behaviour of the function computed by support vector machines (SVM) and related algorithms that minimize a regularized empirical convex loss function in the reproducing kernel Hilbert space of the Gaussian RBF kernel, in the situation where the number of examples tends to inﬁnity, the bandwidth of the Gaussian kernel tends to 0, and the regularization parameter is held ﬁxed. Non-asymptotic convergence bounds to this limit in the L2 sense are provided, together with upper bounds on the classiﬁcation error that is shown to converge to the Bayes risk, therefore proving the Bayes-consistency of a variety of methods although the regularization term does not vanish. These results are particularly relevant to the one-class SVM, for which the regularization can not vanish by construction, and which is shown for the ﬁrst time to be a consistent density level set estimator.},
	language = {en},
	author = {Vert, Regis and Vert, Jean-Philippe},
}

@misc{bonnier_kernelized_2023,
	title = {Kernelized {Cumulants}: {Beyond} {Kernel} {Mean} {Embeddings}},
	shorttitle = {Kernelized {Cumulants}},
	url = {http://arxiv.org/abs/2301.12466},
	abstract = {In \${\textbackslash}mathbb R{\textasciicircum}d\$, it is well-known that cumulants provide an alternative to moments that can achieve the same goals with numerous benefits such as lower variance estimators. In this paper we extend cumulants to reproducing kernel Hilbert spaces (RKHS) using tools from tensor algebras and show that they are computationally tractable by a kernel trick. These kernelized cumulants provide a new set of all-purpose statistics; the classical maximum mean discrepancy and Hilbert-Schmidt independence criterion arise as the degree one objects in our general construction. We argue both theoretically and empirically (on synthetic, environmental, and traffic data analysis) that going beyond degree one has several advantages and can be achieved with the same computational complexity and minimal overhead in our experiments.},
	urldate = {2023-09-21},
	publisher = {arXiv},
	author = {Bonnier, Patric and Oberhauser, Harald and Szabó, Zoltán},
	month = jan,
	year = {2023},
	note = {arXiv:2301.12466 [cs, math, stat]},
	keywords = {Computer Science - Information Theory, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{lin_causal_2022-1,
	title = {Causal {Inference} on {Distribution} {Functions}},
	url = {http://arxiv.org/abs/2101.01599},
	abstract = {Understanding causal relationships is one of the most important goals of modern science. So far, the causal inference literature has focused almost exclusively on outcomes coming from the Euclidean space \${\textbackslash}mathbb\{R\}{\textasciicircum}p\$. However, it is increasingly common that complex datasets are best summarized as data points in non-linear spaces. In this paper, we present a novel framework of causal effects for outcomes from the Wasserstein space of cumulative distribution functions, which in contrast to the Euclidean space, is non-linear. We develop doubly robust estimators and associated asymptotic theory for these causal effects. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.},
	urldate = {2023-09-20},
	publisher = {arXiv},
	author = {Lin, Zhenhua and Kong, Dehan and Wang, Linbo},
	month = oct,
	year = {2022},
	note = {arXiv:2101.01599 [stat]},
	keywords = {Statistics - Methodology},
}

@inproceedings{lerasle_monk_2018,
	title = {{MONK} - {Outlier}-{Robust} {Mean} {Embedding} {Estimation} by {Median}-of-{Means}},
	url = {https://www.semanticscholar.org/paper/MONK-Outlier-Robust-Mean-Embedding-Estimation-by-Lerasle-Szab%C3%B3/06e0b453343ef808e0a4699fe60fda8acd9cdaf0},
	abstract = {Mean embeddings provide an extremely flexible and powerful tool in machine learning and statistics to represent probability distributions and define a semi-metric (MMD, maximum mean discrepancy; also called N-distance or energy distance), with numerous successful applications. The representation is constructed as the expectation of the feature map defined by a kernel. As a mean, its classical empirical estimator, however, can be arbitrary severely affected even by a single outlier in case of unbounded features. To the best of our knowledge, unfortunately even the consistency of the existing few techniques trying to alleviate this serious sensitivity bottleneck is unknown. In this paper, we show how the recently emerged principle of median-of-means can be used to design estimators for kernel mean embedding and MMD with excessive resistance properties to outliers, and optimal sub-Gaussian deviation bounds under mild assumptions.},
	urldate = {2023-09-24},
	author = {Lerasle, M. and Szabó, Z. and Lecué, Guillaume and Massiot, Gaspar and Moulines, É},
	month = feb,
	year = {2018},
}

@inproceedings{glazer_q-ocsvm_2013,
	title = {q-{OCSVM}: {A} q-{Quantile} {Estimator} for {High}-{Dimensional} {Distributions}},
	volume = {26},
	shorttitle = {q-{OCSVM}},
	url = {https://proceedings.neurips.cc/paper_files/paper/2013/hash/819f46e52c25763a55cc642422644317-Abstract.html},
	urldate = {2023-09-19},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Glazer, Assaf and Lindenbaum, Michael and Markovitch, Shaul},
	year = {2013},
}

@misc{lin_causal_2022-2,
	title = {Causal {Inference} on {Distribution} {Functions}},
	url = {http://arxiv.org/abs/2101.01599},
	abstract = {Understanding causal relationships is one of the most important goals of modern science. So far, the causal inference literature has focused almost exclusively on outcomes coming from the Euclidean space \${\textbackslash}mathbb\{R\}{\textasciicircum}p\$. However, it is increasingly common that complex datasets are best summarized as data points in non-linear spaces. In this paper, we present a novel framework of causal effects for outcomes from the Wasserstein space of cumulative distribution functions, which in contrast to the Euclidean space, is non-linear. We develop doubly robust estimators and associated asymptotic theory for these causal effects. As an illustration, we use our framework to quantify the causal effect of marriage on physical activity patterns using wearable device data collected through the National Health and Nutrition Examination Survey.},
	urldate = {2023-09-21},
	publisher = {arXiv},
	author = {Lin, Zhenhua and Kong, Dehan and Wang, Linbo},
	month = oct,
	year = {2022},
	note = {arXiv:2101.01599 [stat]},
	keywords = {Statistics - Methodology},
}

@book{coles_introduction_2001,
	address = {London},
	series = {Springer {Series} in {Statistics}},
	title = {An {Introduction} to {Statistical} {Modeling} of {Extreme} {Values}},
	isbn = {978-1-84996-874-4 978-1-4471-3675-0},
	url = {http://link.springer.com/10.1007/978-1-4471-3675-0},
	language = {en},
	urldate = {2023-09-21},
	publisher = {Springer London},
	author = {Coles, Stuart},
	year = {2001},
	doi = {10.1007/978-1-4471-3675-0},
}

@misc{martinez-taboada_efficient_2023,
	title = {An {Efficient} {Doubly}-{Robust} {Test} for the {Kernel} {Treatment} {Effect}},
	url = {http://arxiv.org/abs/2304.13237},
	abstract = {The average treatment effect, which is the difference in expectation of the counterfactuals, is probably the most popular target effect in causal inference with binary treatments. However, treatments may have effects beyond the mean, for instance decreasing or increasing the variance. We propose a new kernel-based test for distributional effects of the treatment. It is, to the best of our knowledge, the first kernel-based, doubly-robust test with provably valid type-I error. Furthermore, our proposed algorithm is efficient, avoiding the use of permutations.},
	urldate = {2023-09-21},
	publisher = {arXiv},
	author = {Martinez-Taboada, Diego and Ramdas, Aaditya and Kennedy, Edward H.},
	month = apr,
	year = {2023},
	note = {arXiv:2304.13237 [stat]},
	keywords = {Statistics - Machine Learning, Statistics - Methodology},
}

@inproceedings{park_conditional_2021,
	title = {Conditional {Distributional} {Treatment} {Effect} with {Kernel} {Conditional} {Mean} {Embeddings} and {U}-{Statistic} {Regression}},
	url = {https://proceedings.mlr.press/v139/park21c.html},
	abstract = {We propose to analyse the conditional distributional treatment effect (CoDiTE), which, in contrast to the more common conditional average treatment effect (CATE), is designed to encode a treatment’s distributional aspects beyond the mean. We first introduce a formal definition of the CoDiTE associated with a distance function between probability measures. Then we discuss the CoDiTE associated with the maximum mean discrepancy via kernel conditional mean embeddings, which, coupled with a hypothesis test, tells us whether there is any conditional distributional effect of the treatment. Finally, we investigate what kind of conditional distributional effect the treatment has, both in an exploratory manner via the conditional witness function, and in a quantitative manner via U-statistic regression, generalising the CATE to higher-order moments. Experiments on synthetic, semi-synthetic and real datasets demonstrate the merits of our approach.},
	language = {en},
	urldate = {2023-09-21},
	booktitle = {Proceedings of the 38th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Park, Junhyung and Shalit, Uri and Schölkopf, Bernhard and Muandet, Krikamol},
	month = jul,
	year = {2021},
	note = {ISSN: 2640-3498},
	pages = {8401--8412},
}

@inproceedings{hartford_deep_2017,
	title = {Deep {IV}: {A} {Flexible} {Approach} for {Counterfactual} {Prediction}},
	shorttitle = {Deep {IV}},
	url = {https://proceedings.mlr.press/v70/hartford17a.html},
	abstract = {Counterfactual prediction requires understanding causal relationships between so-called treatment and outcome variables. This paper provides a recipe for augmenting deep learning methods to accurately characterize such relationships in the presence of instrument variables (IVs) – sources of treatment randomization that are conditionally independent from the outcomes. Our IV specification resolves into two prediction tasks that can be solved with deep neural nets: a first-stage network for treatment prediction and a second-stage network whose loss function involves integration over the conditional treatment distribution. This Deep IV framework allows us to take advantage of off-the-shelf supervised learning techniques to estimate causal effects by adapting the loss function. Experiments show that it outperforms existing machine learning approaches.},
	language = {en},
	urldate = {2023-11-08},
	booktitle = {Proceedings of the 34th {International} {Conference} on {Machine} {Learning}},
	publisher = {PMLR},
	author = {Hartford, Jason and Lewis, Greg and Leyton-Brown, Kevin and Taddy, Matt},
	month = jul,
	year = {2017},
	pages = {1414--1423},
}

@article{gardenfors_unreliable_1982,
	title = {Unreliable {Probabilities}, {Risk} {Taking}, and {Decision} {Making}},
	volume = {53},
	issn = {0039-7857},
	url = {https://www.jstor.org/stable/20115812},
	number = {3},
	urldate = {2023-11-08},
	journal = {Synthese},
	author = {Gärdenfors, Peter and Sahlin, Nils-Eric},
	year = {1982},
	note = {Publisher: Springer},
	pages = {361--386},
}

@article{bernard_introduction_2005,
	title = {An introduction to the imprecise {Dirichlet} model for multinomial data},
	volume = {39},
	issn = {0888613X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X04001069},
	doi = {10.1016/j.ijar.2004.10.002},
	abstract = {The imprecise Dirichlet model (IDM) was recently proposed by Walley as a model for objective statistical inference from multinomial data with chances h. In the IDM, prior or posterior uncertainty about h is described by a set of Dirichlet distributions, and inferences about events are summarized by lower and upper probabilities. The IDM avoids shortcomings of alternative objective models, either frequentist or Bayesian. We review the properties of the model, for both parametric and predictive inferences, and some of its recent applications to various statistical problems.},
	language = {en},
	number = {2-3},
	urldate = {2023-11-08},
	journal = {International Journal of Approximate Reasoning},
	author = {Bernard, Jean-Marc},
	month = jun,
	year = {2005},
	pages = {123--150},
}

@misc{shaker_aleatoric_2020-1,
	title = {Aleatoric and {Epistemic} {Uncertainty} with {Random} {Forests}},
	url = {http://arxiv.org/abs/2001.00893},
	abstract = {Due to the steadily increasing relevance of machine learning for practical applications, many of which are coming with safety requirements, the notion of uncertainty has received increasing attention in machine learning research in the last couple of years. In particular, the idea of distinguishing between two important types of uncertainty, often refereed to as aleatoric and epistemic, has recently been studied in the setting of supervised learning. In this paper, we propose to quantify these uncertainties with random forests. More specifically, we show how two general approaches for measuring the learner's aleatoric and epistemic uncertainty in a prediction can be instantiated with decision trees and random forests as learning algorithms in a classification setting. In this regard, we also compare random forests with deep neural networks, which have been used for a similar purpose.},
	urldate = {2023-11-07},
	publisher = {arXiv},
	author = {Shaker, Mohammad Hossein and Hüllermeier, Eyke},
	month = jan,
	year = {2020},
	note = {arXiv:2001.00893 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@book{augustin_introduction_2014,
	address = {Hoboken, NJ},
	series = {Wiley series in probability and statistics},
	title = {Introduction to imprecise probabilities},
	isbn = {978-0-470-97381-3},
	language = {en},
	publisher = {Wiley},
	editor = {Augustin, Thomas and Coolen, Frank P. A. and De Cooman, Gert and Troffaes, Matthias C. M.},
	year = {2014},
}

@misc{lienen_credal_2021,
	title = {Credal {Self}-{Supervised} {Learning}},
	url = {http://arxiv.org/abs/2106.11853},
	abstract = {Self-training is an effective approach to semi-supervised learning. The key idea is to let the learner itself iteratively generate "pseudo-supervision" for unlabeled instances based on its current hypothesis. In combination with consistency regularization, pseudo-labeling has shown promising performance in various domains, for example in computer vision. To account for the hypothetical nature of the pseudo-labels, these are commonly provided in the form of probability distributions. Still, one may argue that even a probability distribution represents an excessive level of informedness, as it suggests that the learner precisely knows the ground-truth conditional probabilities. In our approach, we therefore allow the learner to label instances in the form of credal sets, that is, sets of (candidate) probability distributions. Thanks to this increased expressiveness, the learner is able to represent uncertainty and a lack of knowledge in a more flexible and more faithful manner. To learn from weakly labeled data of that kind, we leverage methods that have recently been proposed in the realm of so-called superset learning. In an exhaustive empirical evaluation, we compare our methodology to state-of-the-art self-supervision approaches, showing competitive to superior performance especially in low-label scenarios incorporating a high degree of uncertainty.},
	urldate = {2023-11-07},
	publisher = {arXiv},
	author = {Lienen, Julian and Hüllermeier, Eyke},
	month = nov,
	year = {2021},
	note = {arXiv:2106.11853 [cs, stat]},
	keywords = {Computer Science - Artificial Intelligence, Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{klein_interpreting_2016,
	title = {Interpreting evidential distances by connecting them to partial orders: {Application} to belief function approximation},
	volume = {71},
	issn = {0888613X},
	shorttitle = {Interpreting evidential distances by connecting them to partial orders},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X16000025},
	doi = {10.1016/j.ijar.2016.01.001},
	language = {en},
	urldate = {2023-11-07},
	journal = {International Journal of Approximate Reasoning},
	author = {Klein, John and Destercke, Sebastien and Colot, Olivier},
	month = apr,
	year = {2016},
	pages = {15--33},
}

@incollection{kacprzyk_bayesian_2012,
	address = {Berlin, Heidelberg},
	title = {Bayesian {Networks} with {Imprecise} {Probabilities}: {Theory} and {Application} to {Classification}},
	volume = {23},
	isbn = {978-3-642-23165-0 978-3-642-23166-7},
	shorttitle = {Bayesian {Networks} with {Imprecise} {Probabilities}},
	url = {http://link.springer.com/10.1007/978-3-642-23166-7_4},
	abstract = {Bayesian network are powerful probabilistic graphical models for modelling uncertainty. Among others, classiﬁcation represents an important application: some of the most used classiﬁers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided for quantiﬁcation. This requirement is sometimes too narrow. Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions. This leads to a novel class of imprecise probabilistic graphical models, called credal networks. In particular, classiﬁers based on Bayesian networks are generalized to so-called credal classiﬁers. Unlike Bayesian classiﬁers, which always detect a single class as the one maximizing the posterior class probability, a credal classiﬁer may eventually be unable to discriminate a single class. In other words, if the available information is not sufﬁcient, credal classiﬁers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian classiﬁers.},
	language = {en},
	urldate = {2023-11-06},
	booktitle = {Data {Mining}: {Foundations} and {Intelligent} {Paradigms}},
	publisher = {Springer Berlin Heidelberg},
	author = {Corani, G. and Antonucci, A. and Zaffalon, M.},
	editor = {Kacprzyk, Janusz and Jain, Lakhmi C. and Holmes, Dawn E. and Jain, Lakhmi C.},
	year = {2012},
	doi = {10.1007/978-3-642-23166-7_4},
	note = {Series Title: Intelligent Systems Reference Library},
	pages = {49--93},
}

@article{hullermeier_quantication_nodate,
	title = {Quantiﬁcation of {Credal} {Uncertainty} in {Machine} {Learning}: {A} {Critical} {Analysis} and {Empirical} {Comparison}},
	abstract = {The representation and quantiﬁcation of uncertainty has received increasing attention in machine learning in the recent past. The formalism of credal sets provides an interesting alternative in this regard, especially as it combines the representation of epistemic (lack of knowledge) and aleatoric (statistical) uncertainty in a rather natural way. In this paper, we elaborate on uncertainty measures for credal sets from the perspective of machine learning. More speciﬁcally, we provide an overview of proposals, discuss existing measures in a critical way, and also propose a new measure that is more tailored to the machine learning setting. Based on an experimental study, we conclude that theoretically well-justiﬁed measures also lead to better performance in practice. Besides, we corroborate the difﬁculty of the disaggregation problem, that is, of decomposing the amount of total uncertainty into aleatoric and epistemic uncertainty in a sound manner, thereby complementing theoretical ﬁndings with empirical evidence.},
	language = {en},
	author = {Hüllermeier, Eyke and Destercke, Sebastian and Shaker, Mohammad Hossein},
}

@article{mork_uncertainty_2013-1,
	title = {Uncertainty, credal sets and second order probability},
	volume = {190},
	issn = {0039-7857},
	url = {https://www.jstor.org/stable/23324632},
	abstract = {The last 20 years or so has seen an intense search carried out within Dempster—Shafer theory, with the aim of finding a generalization of the Shannon entropy for belief functions. In that time, there has also been much progress made in credal set theory—another generalization of the traditional Bayesian epistemic representation—albeit not in this particular area. In credal set theory, sets of probability functions are utilized to represent the epistemic state of rational agents instead of the single probability function of traditional Bayesian theory. The Shannon entropy has been shown to uniquely capture certain highly intuitive properties of uncertainty, and can thus be considered a measure of that quantity. This article presents two measures developed with the purpose of generalizing the Shannon entropy for (1) unordered convex credal sets and (2) possibly non-convex credal sets ordered by second order probability, thereby providing uncertainty measures for such epistemic representations. There is also a comparison with the results of the measure AU developed within Dempster—Shafer theory in a few instances where unordered convex credal set theory and Dempster—Shafer theory overlap.},
	number = {3},
	urldate = {2023-11-06},
	journal = {Synthese},
	author = {Mork, Jonas Clausen},
	year = {2013},
	note = {Publisher: Springer},
	pages = {353--378},
}

@misc{hibshman_higher_2021,
	title = {Higher {Order} {Imprecise} {Probabilities} and {Statistical} {Testing}},
	url = {http://arxiv.org/abs/2107.04542},
	abstract = {We generalize standard credal set models for imprecise probabilities to include higher order credal sets -- confidences about confidences. In doing so, we specify how an agent's higher order confidences (credal sets) update upon observing an event. Our model begins to address standard issues with imprecise probability models, like Dilation and Belief Inertia. We conjecture that when higher order credal sets contain all possible probability functions, then in the limiting case the highest order confidences converge to form a uniform distribution over the first order credal set, where we define uniformity in terms of the statistical distance metric (total variation distance). Finite simulation supports the conjecture. We further suggest that this convergence presents the total-variation-uniform distribution as a natural, privileged prior for statistical hypothesis testing.},
	urldate = {2023-11-06},
	publisher = {arXiv},
	author = {Hibshman, Justus and Weninger, Tim},
	month = jul,
	year = {2021},
	note = {arXiv:2107.04542 [math, stat]},
	keywords = {Mathematics - Statistics Theory, Statistics - Methodology},
}

@article{bronevich_modelling_2018-2,
	title = {Modelling uncertainty with generalized credal sets: application to conjunction and decision},
	volume = {47},
	issn = {0308-1079, 1563-5104},
	shorttitle = {Modelling uncertainty with generalized credal sets},
	url = {https://www.tandfonline.com/doi/full/10.1080/03081079.2017.1391805},
	doi = {10.1080/03081079.2017.1391805},
	language = {en},
	number = {1},
	urldate = {2023-11-06},
	journal = {International Journal of General Systems},
	author = {Bronevich, Andrey G. and Rozenberg, Igor N.},
	month = jan,
	year = {2018},
	pages = {67--96},
}

@article{bronevich_characteristics_2017,
	title = {Some {Characteristics} of {Credal} {Sets} and {Their} {Application} to {Analysis} of {Polls} {Results}},
	volume = {122},
	issn = {18770509},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S1877050917326534},
	doi = {10.1016/j.procs.2017.11.408},
	language = {en},
	urldate = {2023-11-06},
	journal = {Procedia Computer Science},
	author = {Bronevich, Andrey G. and Spiridenkova, Natalia S.},
	year = {2017},
	pages = {572--578},
}

@article{tretiak_neural_2023,
	title = {Neural network model for imprecise regression with interval dependent variables},
	volume = {161},
	issn = {08936080},
	url = {http://arxiv.org/abs/2206.02467},
	doi = {10.1016/j.neunet.2023.02.005},
	abstract = {This paper presents a computationally feasible method to compute rigorous bounds on the interval-generalisation of regression analysis to account for epistemic uncertainty in the output variables. The new iterative method uses machine learning algorithms to fit an imprecise regression model to data that consist of intervals rather than point values. The method is based on a single-layer interval neural network which can be trained to produce an interval prediction. It seeks parameters for the optimal model that minimizes the mean squared error between the actual and predicted interval values of the dependent variable using a first-order gradient-based optimization and interval analysis computations to model the measurement imprecision of the data. An additional extension to a multi-layer neural network is also presented. We consider the explanatory variables to be precise point values, but the measured dependent values are characterized by interval bounds without any probabilistic information. The proposed iterative method estimates the lower and upper bounds of the expectation region, which is an envelope of all possible precise regression lines obtained by ordinary regression analysis based on any configuration of real-valued points from the respective y-intervals and their x-values.},
	urldate = {2023-09-12},
	journal = {Neural Networks},
	author = {Tretiak, Krasymyr and Schollmeyer, Georg and Ferson, Scott},
	month = apr,
	year = {2023},
	note = {arXiv:2206.02467 [physics, stat]},
	keywords = {68T07, 62J02, 62J05, Mathematics - Statistics Theory, Physics - Data Analysis, Statistics and Probability},
	pages = {550--564},
}

@inproceedings{miranda_unifying_2019,
	title = {A {Unifying} {Frame} for {Neighbourhood} and {Distortion} {Models}},
	url = {https://proceedings.mlr.press/v103/miranda19a.html},
	abstract = {Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari-mutuel model, linear vacuous mixtures or the constant odds ratio model. In this paper, we show that all of them can be represented as probability sets that are neighbourhoods defined over different (pre)-metrics, providing a unified view of such models. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, … thus providing possible guidelines to pick a neighbourhood rather than another.},
	language = {en},
	urldate = {2023-09-04},
	booktitle = {Proceedings of the {Eleventh} {International} {Symposium} on {Imprecise} {Probabilities}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Miranda, Enrique and Montes, Ignacio and Destercke, Sébastien},
	month = jun,
	year = {2019},
	note = {ISSN: 2640-3498},
	pages = {304--313},
}

@inproceedings{montes_neighbourhood_2023,
	title = {Neighbourhood models induced by the {Euclidean} distance and the {Kullback}-{Leibler} divergence},
	url = {https://proceedings.mlr.press/v215/montes23a.html},
	abstract = {Neighbourhood or distortion models are particular imprecise probability models that appear by creating a neighbourhood around a probability measure given a distorting function and a distortion parameter. This paper investigates the distortion models obtained when considering the Euclidean distance or the Kullback-Leibler divergence as distorting function. We analyse the main properties of the credal sets induced by these two distorting functions as well as the main properties of the associated coherent lower previsions. To conclude the paper, we compare these two models with other well-known distortion models: the pari-mutuel, linear vacuous, constant odds ratio and total variation models.},
	language = {en},
	urldate = {2023-09-11},
	booktitle = {Proceedings of the {Thirteenth} {International} {Symposium} on {Imprecise} {Probability}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Montes, Ignacio},
	month = jul,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {367--378},
}

@incollection{agresti_naive_2012,
	address = {New York, NY},
	title = {Naive {Probabilism}},
	isbn = {978-1-4614-3648-5 978-1-4614-3649-2},
	url = {https://link.springer.com/10.1007/978-1-4614-3649-2_18},
	language = {en},
	urldate = {2023-08-06},
	booktitle = {Strength in {Numbers}: {The} {Rising} of {Academic} {Statistics} {Departments} in the {U}. {S}.},
	publisher = {Springer New York},
	author = {Naus, Joseph},
	editor = {Agresti, Alan and Meng, Xiao-Li},
	year = {2012},
	doi = {10.1007/978-1-4614-3649-2_18},
	pages = {243--256},
}

@article{rahimian_distributionally_2022,
	title = {Distributionally {Robust} {Optimization}: {A} {Review}},
	volume = {3},
	issn = {2777-5860},
	shorttitle = {Distributionally {Robust} {Optimization}},
	url = {http://arxiv.org/abs/1908.05659},
	doi = {10.5802/ojmo.15},
	abstract = {The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed signiﬁcantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received signiﬁcant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization.},
	language = {en},
	urldate = {2023-04-21},
	journal = {Open Journal of Mathematical Optimization},
	author = {Rahimian, Hamed and Mehrotra, Sanjay},
	month = jul,
	year = {2022},
	note = {arXiv:1908.05659 [cs, math, stat]},
	keywords = {Computer Science - Machine Learning, Mathematics - Optimization and Control, Statistics - Machine Learning},
	pages = {1--85},
}

@article{hullermeier_quantication_nodate,
	title = {Quantiﬁcation of {Credal} {Uncertainty} in {Machine} {Learning}: {A} {Critical} {Analysis} and {Empirical} {Comparison}},
	abstract = {The representation and quantiﬁcation of uncertainty has received increasing attention in machine learning in the recent past. The formalism of credal sets provides an interesting alternative in this regard, especially as it combines the representation of epistemic (lack of knowledge) and aleatoric (statistical) uncertainty in a rather natural way. In this paper, we elaborate on uncertainty measures for credal sets from the perspective of machine learning. More speciﬁcally, we provide an overview of proposals, discuss existing measures in a critical way, and also propose a new measure that is more tailored to the machine learning setting. Based on an experimental study, we conclude that theoretically well-justiﬁed measures also lead to better performance in practice. Besides, we corroborate the difﬁculty of the disaggregation problem, that is, of decomposing the amount of total uncertainty into aleatoric and epistemic uncertainty in a sound manner, thereby complementing theoretical ﬁndings with empirical evidence.},
	language = {en},
	author = {Hüllermeier, Eyke and Destercke, Sebastian and Shaker, Mohammad Hossein},
}

@article{jiao_distributed_nodate,
	title = {Distributed {Distributionally} {Robust} {Optimization} with {Non}-{Convex} {Objectives}},
	abstract = {Distributionally Robust Optimization (DRO), which aims to find an optimal decision that minimizes the worst case cost over the ambiguity set of probability distribution, has been widely applied in diverse applications, e.g., network behavior analysis, risk management, etc. However, existing DRO techniques face three key challenges: 1) how to deal with the asynchronous updating in a distributed environment; 2) how to leverage the prior distribution effectively; 3) how to properly adjust the degree of robustness according to different scenarios. To this end, we propose an asynchronous distributed algorithm, named Asynchronous Single-looP alternatIve gRadient projEction (ASPIRE) algorithm with the itErative Active SEt method (EASE) to tackle the distributed distributionally robust optimization (DDRO) problem. Furthermore, a new uncertainty set, i.e., constrained D-norm uncertainty set, is developed to effectively leverage the prior distribution and flexibly control the degree of robustness. Finally, our theoretical analysis elucidates that the proposed algorithm is guaranteed to converge and the iteration complexity is also analyzed. Extensive empirical studies on real-world datasets demonstrate that the proposed method can not only achieve fast convergence, and remain robust against data heterogeneity as well as malicious attacks, but also tradeoff robustness with performance.},
	language = {en},
	author = {Jiao, Yang and Yang, Kai and Song, Dongjin},
}

@article{guillaume_distributionally_2023,
	title = {Distributionally robust possibilistic optimization problems},
	volume = {454},
	issn = {01650114},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0165011422001877},
	doi = {10.1016/j.fss.2022.05.007},
	abstract = {In this paper a class of optimization problems with uncertain linear constraints is discussed. It is assumed that the constraint coefﬁcients are random vectors whose probability distributions are only partially known. Possibility theory is used to model imprecise probabilities. In one of interpretation, a possibility distribution (a membership function of a fuzzy set) in the set of coefﬁcient realizations induces a necessity measure, which in turn deﬁnes a family of probability distributions in this set. The distributionally robust approach is then used to transform the imprecise constraints into deterministic counterparts. Namely, the uncertain left-hand side of each constraint is replaced with the expected value with respect to the worst probability distribution that can occur. It is shown how to represent the resulting problem by using linear or second-order cone constraints. This leads to problems which are computationally tractable for a wide class of optimization models, in particular for linear programming.},
	language = {en},
	urldate = {2023-09-25},
	journal = {Fuzzy Sets and Systems},
	author = {Guillaume, Romain and Kasperski, Adam and Zieliński, Paweł},
	month = feb,
	year = {2023},
	pages = {56--73},
}

@misc{vovk_logic_2023,
	title = {Logic of subjective probability},
	url = {http://arxiv.org/abs/2309.01173},
	abstract = {In this paper I discuss both syntax and semantics of subjective probability. The semantics determines ways of testing probability statements. Among important varieties of subjective probabilities are intersubjective probabilities and impersonal probabilities, and I will argue that well-tested impersonal probabilities acquire features of objective probabilities. Jeffreys's law, my next topic, states that two successful probability forecasters must issue forecasts that are close to each other, thus supporting the idea of objective probabilities. Finally, I will discuss connections between subjective and frequentist probability.},
	urldate = {2023-09-26},
	publisher = {arXiv},
	author = {Vovk, Vladimir},
	month = sep,
	year = {2023},
	note = {arXiv:2309.01173 [cs, stat]},
	keywords = {62A01 (Primary) 60A99, 62F15, 60F15, 62C05, 60G42, 62M20 (Secondary), Computer Science - Artificial Intelligence, Statistics - Other Statistics},
}

@misc{kannan_residuals-based_2022,
	title = {Residuals-based distributionally robust optimization with covariate information},
	url = {http://arxiv.org/abs/2012.01088},
	abstract = {We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.},
	urldate = {2023-09-25},
	publisher = {arXiv},
	author = {Kannan, Rohit and Bayraksan, Güzin and Luedtke, James R.},
	month = may,
	year = {2022},
	note = {arXiv:2012.01088 [math, stat]},
	keywords = {90C15, 90C47, Mathematics - Optimization and Control, Statistics - Machine Learning},
}

@incollection{hajek_interpretations_2019,
	edition = {Fall 2019},
	title = {Interpretations of {Probability}},
	url = {https://plato.stanford.edu/archives/fall2019/entries/probability-interpret/},
	abstract = {‘The Democrats will probably win the next election.’},
	urldate = {2023-09-29},
	booktitle = {The {Stanford} {Encyclopedia} of {Philosophy}},
	publisher = {Metaphysics Research Lab, Stanford University},
	author = {Hájek, Alan},
	editor = {Zalta, Edward N.},
	year = {2019},
	keywords = {Carnap, Rudolf, Laplace, Pierre Simon, Popper, Karl, Ramsey, Frank, Reichenbach, Hans, causal models, causation: probabilistic, chance: versus randomness, disagreement, epistemology: Bayesian, information, probability, in medieval and Renaissance philosophy, quantum theory: philosophical issues in, statistics, philosophy of},
}

@inproceedings{lee_minimax_2018,
	title = {Minimax {Statistical} {Learning} with {Wasserstein} distances},
	volume = {31},
	url = {https://proceedings.neurips.cc/paper_files/paper/2018/hash/ea8fcd92d59581717e06eb187f10666d-Abstract.html},
	abstract = {As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wasserstein distance between the source and target domain distributions can be reliably estimated from unlabeled samples.},
	urldate = {2023-10-01},
	booktitle = {Advances in {Neural} {Information} {Processing} {Systems}},
	publisher = {Curran Associates, Inc.},
	author = {Lee, Jaeho and Raginsky, Maxim},
	year = {2018},
}

@article{sun_kernel_2023,
	title = {Kernel {Robust} {Hypothesis} {Testing}},
	volume = {69},
	issn = {0018-9448, 1557-9654},
	url = {https://ieeexplore.ieee.org/document/10103918/},
	doi = {10.1109/TIT.2023.3268207},
	abstract = {The problem of robust hypothesis testing is studied, where under the null and the alternative hypotheses, the data-generating distributions are assumed to be in some uncertainty sets, and the goal is to design a test that performs well under the worst-case distributions over the uncertainty sets. In this paper, uncertainty sets are constructed in a data-driven manner using kernel method, i.e., they are centered around empirical distributions of training samples from the null and alternative hypotheses, respectively; and are constrained via the distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space, i.e., maximum mean discrepancy (MMD). The Bayesian setting and the Neyman-Pearson setting are investigated. For the Bayesian setting where the goal is to minimize the worst-case error probability, an optimal test is firstly obtained when the alphabet is finite. When the alphabet is infinite, a tractable approximation is proposed to quantify the worst-case average error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A direct robust kernel test is also proposed and proved to be exponentially consistent. For the Neyman-Pearson setting, where the goal is to minimize the worst-case probability of miss detection subject to a constraint on the worst-case probability of false alarm, an efficient robust kernel test is proposed and is shown to be asymptotically optimal. Numerical results are provided to demonstrate the performance of the proposed robust tests.},
	language = {en},
	number = {10},
	urldate = {2023-10-02},
	journal = {IEEE Transactions on Information Theory},
	author = {Sun, Zhongchang and Zou, Shaofeng},
	month = oct,
	year = {2023},
	pages = {6619--6638},
}

@misc{cuzzolin_uncertainty_2021,
	title = {Uncertainty measures: {The} big picture},
	shorttitle = {Uncertainty measures},
	url = {http://arxiv.org/abs/2104.06839},
	abstract = {Probability theory is far from being the most general mathematical theory of uncertainty. A number of arguments point at its inability to describe second-order ('Knightian') uncertainty. In response, a wide array of theories of uncertainty have been proposed, many of them generalisations of classical probability. As we show here, such frameworks can be organised into clusters sharing a common rationale, exhibit complex links, and are characterised by different levels of generality. Our goal is a critical appraisal of the current landscape in uncertainty theory.},
	urldate = {2023-10-03},
	publisher = {arXiv},
	author = {Cuzzolin, Fabio},
	month = apr,
	year = {2021},
	note = {arXiv:2104.06839 [cs, math, stat]},
	keywords = {62A01, 62A86, 60A05, 60A86, 60D05, 68T37, Computer Science - Artificial Intelligence, Mathematics - Probability, Mathematics - Statistics Theory},
}

@article{hullermeier_fuzzy_2005,
	title = {Fuzzy methods in machine learning and data mining: {Status} and prospects},
	volume = {156},
	issn = {01650114},
	shorttitle = {Fuzzy methods in machine learning and data mining},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0165011405002861},
	doi = {10.1016/j.fss.2005.05.036},
	abstract = {Over the past years, methods for the automated induction of models and the extraction of interesting patterns from empirical data have attracted considerable attention in the fuzzy set community. This paper brieﬂy reviews some typical applications and highlights potential contributions that fuzzy set theory can make to machine learning, data mining, and related ﬁelds. The paper concludes with a critical consideration of recent developments and some suggestions for future research directions.},
	language = {en},
	number = {3},
	urldate = {2023-10-09},
	journal = {Fuzzy Sets and Systems},
	author = {Hüllermeier, Eyke},
	month = dec,
	year = {2005},
	pages = {387--406},
}

@article{hullermeier_possibilistic_2003,
	title = {Possibilistic instance-based learning},
	volume = {148},
	issn = {00043702},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0004370203000195},
	doi = {10.1016/S0004-3702(03)00019-5},
	abstract = {A method of instance-based learning is introduced which makes use of possibility theory and fuzzy sets. Particularly, a possibilistic version of the similarity-guided extrapolation principle underlying the instance-based learning paradigm is proposed. This version is compared to the commonly used probabilistic approach from a methodological point of view. Moreover, aspects of knowledge representation such as the modeling of uncertainty are discussed. Taking the possibilistic extrapolation principle as a point of departure, an instance-based learning procedure is outlined which includes the handling of incomplete information, methods for reducing storage requirements and the adaptation of the inﬂuence of stored cases according to their typicality. First theoretical and experimental results showing the efﬁciency of possibilistic instance-based learning are presented as well.},
	language = {en},
	number = {1-2},
	urldate = {2023-10-14},
	journal = {Artificial Intelligence},
	author = {Hüllermeier, Eyke},
	month = aug,
	year = {2003},
	pages = {335--383},
}

@article{krishnapuram_possibilistic_1993,
	title = {A possibilistic approach to clustering},
	volume = {1},
	issn = {10636706},
	url = {http://ieeexplore.ieee.org/document/227387/},
	doi = {10.1109/91.227387},
	language = {en},
	number = {2},
	urldate = {2023-10-15},
	journal = {IEEE Transactions on Fuzzy Systems},
	author = {Krishnapuram, R. and Keller, J.M.},
	month = may,
	year = {1993},
	pages = {98--110},
}

@article{hao_possibilistic_2022,
	title = {Possibilistic classification by support vector networks},
	volume = {149},
	issn = {08936080},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0893608022000405},
	doi = {10.1016/j.neunet.2022.02.007},
	abstract = {In many real-world classification problems, the available information is often uncertain. In order to effectively describe the inherent vagueness and improve the classification performance, this paper proposes a novel possibilistic classification algorithm using support vector machines (SVMs). Based on possibility theory, the proposed algorithm aims at finding a maximal-margin fuzzy hyperplane by solving a fuzzy mathematical optimization problem Moreover, the decision function of the proposed approach is generalized such that the values assigned to the data vectors fall within a specified range and indicate the membership grade of these data vectors in the positive class. The proposed algorithm retains the advantages of fuzzy set theory and SVM theory. The proposed approach is more robust for handling data corrupted by outliers. Moreover, the structural risk minimization principle of SVMs enables the proposed approach to effectively classify the unseen data. Furthermore, the proposed algorithm has additional advantage of using vagueness parameter v for controlling the bounds on fractions of support vectors and errors. The extensive experiments performed on benchmark datasets and real applications demonstrate that the proposed algorithm has satisfactory generalization accuracy and better describes the inherent vagueness in the given dataset.},
	language = {en},
	urldate = {2023-10-15},
	journal = {Neural Networks},
	author = {Hao, Pei-Yi and Chiang, Jung-Hsien and Chen, Yu-De},
	month = may,
	year = {2022},
	pages = {40--56},
}

@article{hullermeier_aleatoric_2021,
	title = {Aleatoric and {Epistemic} {Uncertainty} in {Machine} {Learning}: {An} {Introduction} to {Concepts} and {Methods}},
	volume = {110},
	issn = {0885-6125, 1573-0565},
	shorttitle = {Aleatoric and {Epistemic} {Uncertainty} in {Machine} {Learning}},
	url = {http://arxiv.org/abs/1910.09457},
	doi = {10.1007/s10994-021-05946-3},
	abstract = {The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.},
	number = {3},
	urldate = {2023-03-17},
	journal = {Machine Learning},
	author = {Hüllermeier, Eyke and Waegeman, Willem},
	month = mar,
	year = {2021},
	note = {arXiv:1910.09457 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
	pages = {457--506},
}

@misc{muandet_domain_2013,
	title = {Domain {Generalization} via {Invariant} {Feature} {Representation}},
	url = {http://arxiv.org/abs/1301.2115},
	abstract = {This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the dissimilarity across domains, whilst preserving the functional relationship between input and output variables. A learning-theoretic analysis shows that reducing dissimilarity improves the expected generalization ability of classifiers on new domains, motivating the proposed algorithm. Experimental results on synthetic and real-world datasets demonstrate that DICA successfully learns invariant features and improves classifier performance in practice.},
	urldate = {2023-10-16},
	publisher = {arXiv},
	author = {Muandet, Krikamol and Balduzzi, David and Schölkopf, Bernhard},
	month = jan,
	year = {2013},
	note = {arXiv:1301.2115 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{bernard_introduction_2005,
	title = {An introduction to the imprecise {Dirichlet} model for multinomial data},
	volume = {39},
	issn = {0888613X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X04001069},
	doi = {10.1016/j.ijar.2004.10.002},
	abstract = {The imprecise Dirichlet model (IDM) was recently proposed by Walley as a model for objective statistical inference from multinomial data with chances h. In the IDM, prior or posterior uncertainty about h is described by a set of Dirichlet distributions, and inferences about events are summarized by lower and upper probabilities. The IDM avoids shortcomings of alternative objective models, either frequentist or Bayesian. We review the properties of the model, for both parametric and predictive inferences, and some of its recent applications to various statistical problems.},
	language = {en},
	number = {2-3},
	urldate = {2023-10-17},
	journal = {International Journal of Approximate Reasoning},
	author = {Bernard, Jean-Marc},
	month = jun,
	year = {2005},
	pages = {123--150},
}

@article{senge_reliable_2014,
	title = {Reliable classification: {Learning} classifiers that distinguish aleatoric and epistemic uncertainty},
	volume = {255},
	issn = {00200255},
	shorttitle = {Reliable classification},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0020025513005410},
	doi = {10.1016/j.ins.2013.07.030},
	abstract = {A proper representation of the uncertainty involved in a prediction is an important prerequisite for the acceptance of machine learning and decision support technology in safety–critical application domains such as medical diagnosis. Despite the existence of various probabilistic approaches in these ﬁelds, there is arguably no method that is able to distinguish between two very different sources of uncertainty: aleatoric uncertainty, which is due to statistical variability and effects that are inherently random, and epistemic uncertainty which is caused by a lack of knowledge. In this paper, we propose a method for binary classiﬁcation that does not only produce a prediction of the class of a query instance but also a quantiﬁcation of the two aforementioned sources of uncertainty. Despite being grounded in probability and statistics, the method is formalized within the framework of fuzzy preference relations. The usefulness and reasonableness of our approach is conﬁrmed on a suitable data set with information about patients suffering from chest pain.},
	language = {en},
	urldate = {2023-10-17},
	journal = {Information Sciences},
	author = {Senge, Robin and Bösner, Stefan and Dembczyński, Krzysztof and Haasenritter, Jörg and Hirsch, Oliver and Donner-Banzhoff, Norbert and Hüllermeier, Eyke},
	month = jan,
	year = {2014},
	pages = {16--29},
}

@book{weirich_rational_2021,
	edition = {1},
	title = {Rational {Choice} {Using} {Imprecise} {Probabilities} and {Utilities}},
	isbn = {978-1-108-58220-9 978-1-108-71350-4},
	url = {https://www.cambridge.org/core/product/identifier/9781108582209/type/element},
	abstract = {An agent often does not have precise probabilities or utilities to guide resolution of a decision problem. I advance a principle of rationality for making decisions in such cases. To begin, I represent the doxastic and conative state of an agent with a set of pairs of a probability assignment and a utility assignment. Then I support a decision principle that allows any act that maximizes expected utility according to some pair of assignments in the set. Assuming that computation of an option’s expected utility uses comprehensive possible outcomes that include the option’s risk, no consideration supports a stricter requirement.},
	language = {en},
	urldate = {2023-10-17},
	publisher = {Cambridge University Press},
	author = {Weirich, Paul},
	month = feb,
	year = {2021},
	doi = {10.1017/9781108582209},
}

@article{lassiter_representing_2020,
	title = {Representing credal imprecision: from sets of measures to hierarchical {Bayesian} models},
	volume = {177},
	issn = {0031-8116, 1573-0883},
	shorttitle = {Representing credal imprecision},
	url = {http://link.springer.com/10.1007/s11098-019-01262-8},
	doi = {10.1007/s11098-019-01262-8},
	abstract = {The basic Bayesian model of credence states, where each individual’s belief state is represented by a single probability measure, has been criticized as psychologically implausible, unable to represent the intuitive distinction between precise and imprecise probabilities, and normatively unjustiﬁable due to a need to adopt arbitrary, unmotivated priors. These arguments are often used to motivate a model on which imprecise credal states are represented by sets of probability measures. I connect this debate with recent work in Bayesian cognitive science, where probabilistic models are typically provided with explicit hierarchical structure. Hierarchical Bayesian models are immune to many classic arguments against single-measure models. They represent grades of imprecision in probability assignments automatically, have strong psychological motivation, and can be normatively justiﬁed even when certain arbitrary decisions are required. In addition, hierarchical models show much more plausible learning behavior than ﬂat representations in terms of sets of measures, which—on standard assumptions about update—rule out simple cases of learning from a starting point of total ignorance.},
	language = {en},
	number = {6},
	urldate = {2023-10-18},
	journal = {Philosophical Studies},
	author = {Lassiter, Daniel},
	month = jun,
	year = {2020},
	pages = {1463--1485},
}

@article{cella_valid_2022,
	title = {Valid inferential models for prediction in supervised learning problems},
	volume = {150},
	issn = {0888613X},
	url = {http://arxiv.org/abs/2112.10234},
	doi = {10.1016/j.ijar.2022.08.001},
	abstract = {Prediction, where observed data is used to quantify uncertainty about a future observation, is a fundamental problem in statistics. Prediction sets with coverage probability guarantees are a common solution, but these do not provide probabilistic uncertainty quantiﬁcation in the sense of assigning beliefs to relevant assertions about the future observable. Alternatively, we recommend the use of a probabilistic predictor, a data-dependent (imprecise) probability distribution for the to-bepredicted observation given the observed data. It is essential that the probabilistic predictor be reliable or valid, and here we oﬀer a notion of validity and explore its behavioral and statistical implications. In particular, we show that valid probabilistic predictors avoid sure loss and lead to prediction procedures with desirable frequentist error rate control properties. We also provide a general inferential model construction that yields a provably valid probabilistic predictor, and we illustrate this construction in regression and classiﬁcation applications.},
	language = {en},
	urldate = {2023-10-18},
	journal = {International Journal of Approximate Reasoning},
	author = {Cella, Leonardo and Martin, Ryan},
	month = nov,
	year = {2022},
	note = {arXiv:2112.10234 [math, stat]},
	keywords = {Mathematics - Statistics Theory, Statistics - Methodology},
	pages = {1--18},
}

@inproceedings{martin_fiducial_2023,
	title = {Fiducial inference viewed through a possibility-theoretic inferential model lens},
	url = {https://proceedings.mlr.press/v215/martin23a.html},
	abstract = {Fisher’s fiducial argument is widely viewed as a failed version of Neyman’s theory of confidence limits. But Fisher’s goal—Bayesian-like probabilistic uncertainty quantification without priors—was more ambitious than Neyman’s, and it’s not out of reach. I’ve recently shown that reliable, prior-free probabilistic uncertainty quantification must be grounded in the theory of imprecise probability, and I’ve put forward a possibility-theoretic solution that achieves it. This has been met with resistance, however, in part due to the statistical community’s singular focus on confidence limits. Indeed, if imprecision isn’t needed to answer confidence-limit-related questions, then what’s the point? In this paper, for a class of practically useful models, I explain specifically why the fiducial argument gives valid confidence limits, i.e., it’s the “best probabilistic approximation” of the possibilistic solution I recently advanced. This sheds new light on what the fiducial argument is doing and on what’s lost in terms of reliability when imprecision is ignored and the fiducial argument is pushed for more than just confidence limits.},
	language = {en},
	urldate = {2023-10-18},
	booktitle = {Proceedings of the {Thirteenth} {International} {Symposium} on {Imprecise} {Probability}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Martin, Ryan},
	month = jul,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {299--310},
}

@article{schulte_proceedings_nodate,
	title = {Proceedings - 31. {Workshop} {Computational} {Intelligence} : {Berlin}, 25. - 26. {November} 2021},
	language = {en},
	author = {Schulte, Horst and Hoffmann, Frank and Mikut, Ralf},
}

@article{bengs_pitfalls_nodate,
	title = {Pitfalls of {Epistemic} {Uncertainty} {Quantification} through {Loss} {Minimisation}},
	abstract = {Uncertainty quantification has received increasing attention in machine learning in the recent past. In particular, a distinction between aleatoric and epistemic uncertainty has been found useful in this regard. The latter refers to the learner’s (lack of) knowledge and appears to be especially difficult to measure and quantify. In this paper, we analyse a recent proposal based on the idea of a second-order learner, which yields predictions in the form of distributions over probability distributions. While standard (first-order) learners can be trained to predict accurate probabilities, namely by minimising suitable loss functions on sample data, we show that loss minimisation does not work for second-order predictors: The loss functions proposed for inducing such predictors do not incentivise the learner to represent its epistemic uncertainty in a faithful way.},
	language = {en},
	author = {Bengs, Viktor and Waegeman, Willem},
}

@article{hullermeier_does_2015,
	title = {Does machine learning need fuzzy logic?},
	volume = {281},
	issn = {01650114},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0165011415004133},
	doi = {10.1016/j.fss.2015.09.001},
	abstract = {This article is a short position paper in which the author outlines his (necessarily subjective) perception of current research in fuzzy machine learning, that is, the use of formal concepts and mathematical tools from fuzzy sets and fuzzy logic in the ﬁeld of machine learning. The paper starts with a critical appraisal of previous contributions to fuzzy machine learning and ends with a suggestion of some directions for future work.},
	language = {en},
	urldate = {2023-10-20},
	journal = {Fuzzy Sets and Systems},
	author = {Hüllermeier, Eyke},
	month = dec,
	year = {2015},
	pages = {292--299},
}

@article{couso_fuzzy_2019,
	title = {Fuzzy {Sets} in {Data} {Analysis}: {From} {Statistical} {Foundations} to {Machine} {Learning}},
	volume = {14},
	issn = {1556-603X, 1556-6048},
	shorttitle = {Fuzzy {Sets} in {Data} {Analysis}},
	url = {https://ieeexplore.ieee.org/document/8610265/},
	doi = {10.1109/MCI.2018.2881642},
	language = {en},
	number = {1},
	urldate = {2023-10-20},
	journal = {IEEE Computational Intelligence Magazine},
	author = {Couso, Ines and Borgelt, Christian and Hullermeier, Eyke and Kruse, Rudolf},
	month = feb,
	year = {2019},
	pages = {31--44},
}

@inproceedings{hullermeier_quantification_2022,
	title = {Quantification of {Credal} {Uncertainty} in {Machine} {Learning}: {A} {Critical} {Analysis} and {Empirical} {Comparison}},
	shorttitle = {Quantification of {Credal} {Uncertainty} in {Machine} {Learning}},
	url = {https://proceedings.mlr.press/v180/hullermeier22a.html},
	abstract = {The representation and quantification of uncertainty has received increasing attention in machine learning in the recent past. The formalism of credal sets provides an interesting alternative in this regard, especially as it combines the representation of epistemic (lack of knowledge) and aleatoric (statistical) uncertainty in a rather natural way. In this paper, we elaborate on uncertainty measures for credal sets from the perspective of machine learning. More specifically, we provide an overview of proposals, discuss existing measures in a critical way, and also propose a new measure that is more tailored to the machine learning setting. Based on an experimental study, we conclude that theoretically well-justified measures also lead to better performance in practice. Besides, we corroborate the difficulty of the disaggregation problem, that is, of decomposing the amount of total uncertainty into aleatoric and epistemic uncertainty in a sound manner, thereby complementing theoretical findings with empirical evidence.},
	language = {en},
	urldate = {2023-10-20},
	booktitle = {Proceedings of the {Thirty}-{Eighth} {Conference} on {Uncertainty} in {Artificial} {Intelligence}},
	publisher = {PMLR},
	author = {Hüllermeier, Eyke and Destercke, Sébastien and Shaker, Mohammad Hossein},
	month = aug,
	year = {2022},
	note = {ISSN: 2640-3498},
	pages = {548--557},
}

@misc{postels_hidden_2021,
	title = {The {Hidden} {Uncertainty} in a {Neural} {Networks} {Activations}},
	url = {http://arxiv.org/abs/2012.03082},
	abstract = {The distribution of a neural network’s latent representations has been successfully used to detect Out-of-Distribution (OOD) data. Since OOD detection denotes a popular benchmark for epistemic uncertainty estimates, this raises the question of a deeper correlation. This work investigates whether the distribution of latent representations indeed contains information about the uncertainty associated with the predictions of a neural network. Prior work identiﬁes epistemic uncertainty with the surprise, thus the negative log-likelihood, of observing a particular latent representation, which we verify empirically. Moreover, we demonstrate that the output-conditional distribution of hidden representations allows quantifying aleatoric uncertainty via the entropy of the predictive distribution. We analyze epistemic and aleatoric uncertainty inferred from the representations of different layers and conclude with the exciting ﬁnding that the hidden repesentations of a deterministic neural network indeed contain information about its uncertainty. We verify our ﬁndings on both classiﬁcation and regression models.},
	language = {en},
	urldate = {2023-10-20},
	publisher = {arXiv},
	author = {Postels, Janis and Blum, Hermann and Strümpler, Yannick and Cadena, Cesar and Siegwart, Roland and Van Gool, Luc and Tombari, Federico},
	month = feb,
	year = {2021},
	note = {arXiv:2012.03082 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@inproceedings{nguyen_reliable_2018,
	address = {Stockholm, Sweden},
	title = {Reliable {Multi}-class {Classification} based on {Pairwise} {Epistemic} and {Aleatoric} {Uncertainty}},
	isbn = {978-0-9992411-2-7},
	url = {https://www.ijcai.org/proceedings/2018/706},
	doi = {10.24963/ijcai.2018/706},
	abstract = {We propose a method for reliable prediction in multi-class classiﬁcation, where reliability refers to the possibility of partial abstention in cases of uncertainty. More speciﬁcally, we allow for predictions in the form of preorder relations on the set of classes, thereby generalizing the idea of setvalued predictions. Our approach relies on combining learning by pairwise comparison with a recent proposal for modeling uncertainty in classiﬁcation, in which a distinction is made between reducible (a.k.a. epistemic) uncertainty caused by a lack of information and irreducible (a.k.a. aleatoric) uncertainty due to intrinsic randomness. The problem of combining uncertain pairwise predictions into a most plausible preorder is then formalized as an integer programming problem. Experimentally, we show that our method is able to appropriately balance reliability and precision of predictions.},
	language = {en},
	urldate = {2023-10-20},
	booktitle = {Proceedings of the {Twenty}-{Seventh} {International} {Joint} {Conference} on {Artificial} {Intelligence}},
	publisher = {International Joint Conferences on Artificial Intelligence Organization},
	author = {Nguyen, Vu-Linh and Destercke, Sébastien and Masson, Marie-Hélène and Hüllermeier, Eyke},
	month = jul,
	year = {2018},
	pages = {5089--5095},
}

@inproceedings{mortier_calibration_2023,
	title = {On the {Calibration} of {Probabilistic} {Classifier} {Sets}},
	url = {https://proceedings.mlr.press/v206/mortier23a.html},
	abstract = {Multi-class classification methods that produce sets of probabilistic classifiers, such as ensemble learning methods, are able to model aleatoric and epistemic uncertainty. Aleatoric uncertainty is then typically quantified via the Bayes error, and epistemic uncertainty via the size of the set. In this paper, we extend the notion of calibration, which is commonly used to evaluate the validity of the aleatoric uncertainty representation of a single probabilistic classifier, to assess the validity of an epistemic uncertainty representation obtained by sets of probabilistic classifiers. Broadly speaking, we call a set of probabilistic classifiers calibrated if one can find a calibrated convex combination of these classifiers. To evaluate this notion of calibration, we propose a novel nonparametric calibration test that generalizes an existing test for single probabilistic classifiers to the case of sets of probabilistic classifiers. Making use of this test, we empirically show that ensembles of deep neural networks are often not well calibrated.},
	language = {en},
	urldate = {2023-10-20},
	booktitle = {Proceedings of {The} 26th {International} {Conference} on {Artificial} {Intelligence} and {Statistics}},
	publisher = {PMLR},
	author = {Mortier, Thomas and Bengs, Viktor and Hüllermeier, Eyke and Luca, Stijn and Waegeman, Willem},
	month = apr,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {8857--8870},
}

@misc{utkin_imprecise_2021,
	title = {An {Imprecise} {SHAP} as a {Tool} for {Explaining} the {Class} {Probability} {Distributions} under {Limited} {Training} {Data}},
	url = {http://arxiv.org/abs/2106.09111},
	abstract = {One of the most popular methods of the machine learning prediction explanation is the SHapley Additive exPlanations method (SHAP). An imprecise SHAP as a modification of the original SHAP is proposed for cases when the class probability distributions are imprecise and represented by sets of distributions. The first idea behind the imprecise SHAP is a new approach for computing the marginal contribution of a feature, which fulfils the important efficiency property of Shapley values. The second idea is an attempt to consider a general approach to calculating and reducing interval-valued Shapley values, which is similar to the idea of reachable probability intervals in the imprecise probability theory. A simple special implementation of the general approach in the form of linear optimization problems is proposed, which is based on using the Kolmogorov-Smirnov distance and imprecise contamination models. Numerical examples with synthetic and real data illustrate the imprecise SHAP.},
	urldate = {2023-10-20},
	publisher = {arXiv},
	author = {Utkin, Lev V. and Konstantinov, Andrei V. and Vishniakov, Kirill A.},
	month = jun,
	year = {2021},
	note = {arXiv:2106.09111 [cs, stat]},
	keywords = {Computer Science - Machine Learning, Statistics - Machine Learning},
}

@misc{noauthor_expert_nodate,
	title = {Expert {Knowledge} {Elicitation}: {Subjective} but {Scientific}},
	shorttitle = {Expert {Knowledge} {Elicitation}},
	url = {https://www.tandfonline.com/doi/epdf/10.1080/00031305.2018.1518265?needAccess=true},
	language = {en},
	urldate = {2023-10-23},
	note = {ISSN: 0003-1305},
}

@misc{bengs_second-order_2023,
	title = {On {Second}-{Order} {Scoring} {Rules} for {Epistemic} {Uncertainty} {Quantification}},
	url = {http://arxiv.org/abs/2301.12736},
	abstract = {It is well known that accurate probabilistic predictors can be trained through empirical risk minimisation with proper scoring rules as loss functions. While such learners capture so-called aleatoric uncertainty of predictions, various machine learning methods have recently been developed with the goal to let the learner also represent its epistemic uncertainty, i.e., the uncertainty caused by a lack of knowledge and data. An emerging branch of the literature proposes the use of a second-order learner that provides predictions in terms of distributions on probability distributions. However, recent work has revealed serious theoretical shortcomings for second-order predictors based on loss minimisation. In this paper, we generalise these findings and prove a more fundamental result: There seems to be no loss function that provides an incentive for a second-order learner to faithfully represent its epistemic uncertainty in the same manner as proper scoring rules do for standard (first-order) learners. As a main mathematical tool to prove this result, we introduce the generalised notion of second-order scoring rules.},
	urldate = {2023-10-20},
	publisher = {arXiv},
	author = {Bengs, Viktor and Hüllermeier, Eyke and Waegeman, Willem},
	month = jan,
	year = {2023},
	note = {arXiv:2301.12736 [cs, stat]},
	keywords = {68T37 (Primary) 68T30 (Secondary), Computer Science - Machine Learning, Statistics - Machine Learning},
}

@article{ohagan_expert_2019,
	title = {Expert {Knowledge} {Elicitation}: {Subjective} but {Scientific}},
	volume = {73},
	issn = {0003-1305, 1537-2731},
	shorttitle = {Expert {Knowledge} {Elicitation}},
	url = {https://www.tandfonline.com/doi/full/10.1080/00031305.2018.1518265},
	doi = {10.1080/00031305.2018.1518265},
	abstract = {Expert opinion and judgment enter into the practice of statistical inference and decision-making in numerous ways. Indeed, there is essentially no aspect of scientific investigation in which judgment is not required. Judgment is necessarily subjective, but should be made as carefully, as objectively, and as scientifically as possible.},
	language = {en},
	number = {sup1},
	urldate = {2023-10-23},
	journal = {The American Statistician},
	author = {O’Hagan, Anthony},
	month = mar,
	year = {2019},
	pages = {69--81},
}

@inproceedings{frohlich_towards_2023,
	title = {Towards a strictly frequentist theory of imprecise probability},
	url = {https://proceedings.mlr.press/v215/frohlich23a.html},
	abstract = {Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to statistical phenomena that exhibit diverging relative frequencies. In doing so, we develop a close connection with imprecise probability: the cluster points of relative frequencies yield a coherent upper prevision. We show that a natural frequentist definition of conditional probability recovers the generalized Bayes rule. We prove constructively that, for a finite set of elementary events, there exists a sequence for which the cluster points of relative frequencies coincide with a prespecified set, thereby providing strictly frequentist semantics for coherent upper previsions.},
	language = {en},
	urldate = {2023-10-23},
	booktitle = {Proceedings of the {Thirteenth} {International} {Symposium} on {Imprecise} {Probability}: {Theories} and {Applications}},
	publisher = {PMLR},
	author = {Fröhlich, Christian and Derr, Rabanus and Williamson, Robert C.},
	month = jul,
	year = {2023},
	note = {ISSN: 2640-3498},
	pages = {230--240},
}

@article{di_zio_partial_2017,
	title = {Partial identification in statistical matching with misclassification},
	volume = {82},
	issn = {0888613X},
	url = {https://linkinghub.elsevier.com/retrieve/pii/S0888613X16303462},
	doi = {10.1016/j.ijar.2016.12.015},
	language = {en},
	urldate = {2023-11-03},
	journal = {International Journal of Approximate Reasoning},
	author = {Di Zio, Marco and Vantaggi, Barbara},
	month = mar,
	year = {2017},
	pages = {227--241},
}

@article{kiraly_algebraic_nodate,
	title = {Algebraic {Geometric} {Comparison} of {Probability} {Distributions}},
	abstract = {We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of ﬁnding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identiﬁability criterion.},
	language = {en},
	author = {Kiraly, Franz J},
}
