@article{albert1982,
  title = {Mixtures of {{Dirichlet Distributions}} and {{Estimation}} in {{Contingency Tables}}},
  author = {Albert, James H. and Gupta, Arjun K.},
  year = {1982},
  journal = {The Annals of Statistics},
  volume = {10},
  number = {4},
  eprint = {2240729},
  eprinttype = {jstor},
  pages = {1261--1268},
  publisher = {Institute of Mathematical Statistics},
  issn = {0090-5364},
  urldate = {2024-01-07},
  abstract = {Assuming a multinomial sampling model, prior distributions are developed which can accept prior information about symmetry and independence in a two-way contingency table. Bayesian estimates for the cell probabilities are obtained from the posterior distributions which are attractive alternatives to the usual classical estimates when vague prior information about symmetry or independence is available.}
}

@article{albert2002,
  title = {Statistical Mechanics of Complex Networks},
  author = {Albert, R{\'e}ka and Barab{\'a}si, Albert-L{\'a}szl{\'o}},
  year = {2002},
  month = jan,
  journal = {Reviews of Modern Physics},
  volume = {74},
  number = {1},
  pages = {47--97},
  publisher = {American Physical Society},
  doi = {10.1103/RevModPhys.74.47},
  urldate = {2024-02-10},
  abstract = {Complex networks describe a wide range of systems in nature and society. Frequently cited examples include the cell, a network of chemicals linked by chemical reactions, and the Internet, a network of routers and computers connected by physical links. While traditionally these systems have been modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles. This article reviews the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, the authors discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, the emerging theory of evolving networks, and the interplay between topology and the network's robustness against failures and attacks.}
}

@article{barabasi1999,
  title = {Emergence of {{Scaling}} in {{Random Networks}}},
  author = {Barab{\'a}si, Albert-L{\'a}szl{\'o} and Albert, R{\'e}ka},
  year = {1999},
  month = oct,
  journal = {Science},
  volume = {286},
  number = {5439},
  pages = {509--512},
  publisher = {American Association for the Advancement of Science},
  doi = {10.1126/science.286.5439.509},
  urldate = {2024-02-09},
  abstract = {Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.}
}

@article{bengs2022,
  title = {Pitfalls of {{Epistemic Uncertainty Quantification}} through {{Loss Minimisation}}},
  author = {Bengs, Viktor and H{\"u}llermeier, Eyke and Waegeman, Willem},
  year = {2022},
  month = dec,
  journal = {Advances in Neural Information Processing Systems},
  volume = {35},
  pages = {29205--29216},
  urldate = {2023-08-31},
  langid = {english}
}

@inproceedings{bengs2023,
  title = {On {{Second-Order Scoring Rules}} for {{Epistemic Uncertainty Quantification}}},
  booktitle = {Proceedings of the 40th {{International Conference}} on {{Machine Learning}}},
  author = {Bengs, Viktor and H{\"u}llermeier, Eyke and Waegeman, Willem},
  year = {2023},
  month = jul,
  pages = {2078--2091},
  publisher = {PMLR},
  issn = {2640-3498},
  urldate = {2024-01-17},
  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.},
  langid = {english}
}

@inproceedings{bilos2019,
  title = {Uncertainty on {{Asynchronous Time Event Prediction}}},
  booktitle = {Advances in {{Neural Information Processing Systems}}},
  author = {Bilo{\v s}, Marin and Charpentier, Bertrand and G{\"u}nnemann, Stephan},
  year = {2019},
  volume = {32},
  publisher = {Curran Associates, Inc.},
  urldate = {2024-01-05},
  abstract = {Asynchronous event sequences are the basis of many applications throughout different industries. In this work, we tackle the task of predicting the next event (given a history), and how this prediction changes with the passage of time. Since at some time points (e.g. predictions far into the future) we might not be able to predict anything with confidence, capturing uncertainty in the predictions  is crucial. We present two new architectures, WGP-LN and FD-Dir, modelling the evolution of the distribution on the probability simplex with time-dependent logistic normal and Dirichlet distributions. In both cases, the combination of RNNs with either Gaussian process or function decomposition allows to express rich temporal evolution of the distribution parameters, and naturally captures uncertainty. Experiments on class prediction, time prediction and anomaly detection demonstrate the high performances of our models on various datasets compared to other approaches.}
}

@inproceedings{bojchevski2018,
  title = {Deep {{Gaussian Embedding}} of {{Graphs}}: {{Unsupervised Inductive Learning}} via {{Ranking}}},
  shorttitle = {Deep {{Gaussian Embedding}} of {{Graphs}}},
  booktitle = {International {{Conference}} on {{Learning Representations}}},
  author = {Bojchevski, Aleksandar and G{\"u}nnemann, Stephan},
  year = {2018},
  month = feb,
  urldate = {2024-02-06},
  abstract = {Methods that learn representations of nodes in a graph play a critical role in network analysis since they enable many downstream learning tasks. We propose Graph2Gauss - an approach that can efficiently learn versatile node embeddings on large scale (attributed) graphs that show strong performance on tasks such as link prediction and node classification. Unlike most approaches that represent nodes as point vectors in a low-dimensional continuous space, we embed each node as a Gaussian distribution, allowing us to capture uncertainty about the representation. Furthermore, we propose an unsupervised method that handles inductive learning scenarios and is applicable to different types of graphs: plain/attributed, directed/undirected. By leveraging both the network structure and the associated node attributes, we are able to generalize to unseen nodes without additional training. To learn the embeddings we adopt a personalized ranking formulation w.r.t. the node distances that exploits the natural ordering of the nodes imposed by the network structure. Experiments on real world networks demonstrate the high performance of our approach, outperforming state-of-the-art network embedding methods on several different tasks. Additionally, we demonstrate the benefits of modeling uncertainty - by analyzing it we can estimate neighborhood diversity and detect the intrinsic latent dimensionality of a graph.},
  langid = {english}
}

@inproceedings{borovitskiy2021,
  title = {Mat{\'e}rn {{Gaussian Processes}} on {{Graphs}}},
  booktitle = {Proceedings of {{The}} 24th {{International Conference}} on {{Artificial Intelligence}} and {{Statistics}}},
  author = {Borovitskiy, Viacheslav and Azangulov, Iskander and Terenin, Alexander and Mostowsky, Peter and Deisenroth, Marc and Durrande, Nicolas},
  year = {2021},
  month = mar,
  pages = {2593--2601},
  publisher = {PMLR},
  issn = {2640-3498},
  urldate = {2024-02-10},
  abstract = {Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Mat{\'e}rn Gaussian processes---a widely-used model class in the Euclidean setting---to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Mat{\'e}rn Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks.},
  langid = {english}
}

@inproceedings{bronevich2008,
  title = {Axioms for Uncertainty Measures on Belief Functions and Credal Sets},
  booktitle = {{{NAFIPS}} 2008 - 2008 {{Annual Meeting}} of the {{North American Fuzzy Information Processing Society}}},
  author = {Bronevich, Andrey and Klir, George J.},
  year = {2008},
  month = may,
  pages = {1--6},
  doi = {10.1109/NAFIPS.2008.4531311},
  urldate = {2024-02-08},
  abstract = {In this paper we present a system of axioms for total uncertainty measures, which can be equivalently defined for belief functions and credal sets. This system of axioms provides that a measure of total uncertainty coincides with Shannon entropy on the set of probability measures and it is the Hartley measure on the set of 0,1-valued belief measures. We check this system of axioms for the well-known candidates for total uncertainty measures; in particular, the upper entropy obeys all the necessary requirements. Some properties of such measures of total uncertainty allow us to propose ways for disaggregation of these measures into two parts, which correspond to measures of conflict and measures of nonspecificity.},
  keywords = {Chromium,Entropy,Industrial engineering,Mathematics,Measurement uncertainty,Particle measurements,Terminology}
}

@inproceedings{charpentier2020,
  title = {Posterior Network: Uncertainty Estimation without {{OOD}} Samples via Density-Based Pseudo-Counts},
  shorttitle = {Posterior Network},
  booktitle = {Proceedings of the 34th {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Charpentier, Bertrand and Z{\"u}gner, Daniel and G{\"u}nnemann, Stephan},
  year = {2020},
  month = dec,
  series = {{{NIPS}}'20},
  pages = {1356--1367},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2023-09-05},
  abstract = {Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution data and makes the unrealistic assumption that out-of-distribution (OOD) data is known at training time. In this work we propose the Posterior Network (PostNet), which uses Normalizing Flows to predict an individual closed-form posterior distribution over predicted probabilites for any input sample. The posterior distributions learned by PostNet accurately reflect uncertainty for in- and out-of-distribution data -- without requiring access to OOD data at training time. PostNet achieves state-of-the art results in OOD detection and in uncertainty calibration under dataset shifts.},
  isbn = {978-1-71382-954-6}
}

@article{clemen1999,
  title = {Combining {{Probability Distributions From Experts}} in {{Risk Analysis}}},
  author = {Clemen, Robert T. and Winkler, Robert L.},
  year = {1999},
  month = apr,
  journal = {Risk Analysis},
  volume = {19},
  number = {2},
  pages = {187--203},
  issn = {1573-9147},
  doi = {10.1023/A:1006917509560},
  urldate = {2023-09-05},
  abstract = {This paper concerns the combination of experts' probability distributions in risk analysis, discussing a variety of combination methods and attempting to highlight the important conceptual and practical issues to be considered in designing a combination process in practice. The role of experts is important because their judgments can provide valuable information, particularly in view of the limited availability of ``hard data'' regarding many important uncertainties in risk analysis. Because uncertainties are represented in terms of probability distributions in probabilistic risk analysis (PRA), we consider expert information in terms of probability distributions. The motivation for the use of multiple experts is simply the desire to obtain as much information as possible. Combining experts' probability distributions summarizes the accumulated information for risk analysts and decision-makers. Procedures for combining probability distributions are often compartmentalized as mathematical aggregation methods or behavioral approaches, and we discuss both categories. However, an overall aggregation process could involve both mathematical and behavioral aspects, and no single process is best in all circumstances. An understanding of the pros and cons of different methods and the key issues to consider is valuable in the design of a combination process for a specific PRA. The output, a ``combined probability distribution,'' can ideally be viewed as representing a summary of the current state of expert opinion regarding the uncertainty of interest.},
  langid = {english},
  keywords = {Combining probabilities,expert judgment,probability assessment}
}

@incollection{clemen2007,
  title = {Aggregating {{Probability Distributions}}},
  booktitle = {Advances in {{Decision Analysis}}: {{From Foundations}} to {{Applications}}},
  author = {Clemen, Robert T. and Winkler, Robert L.},
  year = {2007},
  month = jul,
  pages = {154--176},
  publisher = {Cambridge University Press},
  doi = {10.1017/CBO9780511611308.010},
  urldate = {2023-09-05},
  abstract = {Advances in Decision Analysis - July 2007},
  langid = {english}
}

@inproceedings{depeweg2018,
  title = {Decomposition of {{Uncertainty}} in {{Bayesian Deep Learning}} for {{Efficient}} and {{Risk-sensitive Learning}}},
  booktitle = {Proceedings of the 35th {{International Conference}} on {{Machine Learning}}},
  author = {Depeweg, Stefan and {Hernandez-Lobato}, Jose-Miguel and {Doshi-Velez}, Finale and Udluft, Steffen},
  year = {2018},
  month = jul,
  pages = {1184--1193},
  publisher = {PMLR},
  issn = {2640-3498},
  urldate = {2024-03-19},
  abstract = {Bayesian neural networks with latent variables are scalable and flexible probabilistic models: they account for uncertainty in the estimation of the network weights and, by making use of latent variables, can capture complex noise patterns in the data. Using these models we show how to perform and utilize a decomposition of uncertainty in aleatoric and epistemic components for decision making purposes. This allows us to successfully identify informative points for active learning of functions with heteroscedastic and bimodal noise. Using the decomposition we further define a novel risk-sensitive criterion for reinforcement learningto identify policies that balance expected cost, model-bias and noise aversion.},
  langid = {english}
}

@inproceedings{duan2024,
  title = {Evidential {{Uncertainty Quantification}}: {{A Variance-Based Perspective}}},
  shorttitle = {Evidential {{Uncertainty Quantification}}},
  booktitle = {Proceedings of the {{IEEE}}/{{CVF Winter Conference}} on {{Applications}} of {{Computer Vision}}},
  author = {Duan, Ruxiao and Caffo, Brian and Bai, Harrison X. and Sair, Haris I. and Jones, Craig},
  year = {2024},
  pages = {2132--2141},
  urldate = {2024-02-08},
  langid = {english}
}

@inproceedings{gasteiger2018,
  title = {Predict Then {{Propagate}}: {{Graph Neural Networks}} Meet {{Personalized PageRank}}},
  shorttitle = {Predict Then {{Propagate}}},
  booktitle = {International {{Conference}} on {{Learning Representations}}},
  author = {Gasteiger, Johannes and Bojchevski, Aleksandar and G{\"u}nnemann, Stephan},
  year = {2018},
  month = sep,
  urldate = {2024-02-06},
  abstract = {Neural message passing algorithms for semi-supervised classification on graphs have recently achieved great success. However, for classifying a node these methods only consider nodes that are a few propagation steps away and the size of this utilized neighborhood is hard to extend. In this paper, we use the relationship between graph convolutional networks (GCN) and PageRank to derive an improved propagation scheme based on personalized PageRank. We utilize this propagation procedure to construct a simple model, personalized propagation of neural predictions (PPNP), and its fast approximation, APPNP. Our model's training time is on par or faster and its number of parameters on par or lower than previous models. It leverages a large, adjustable neighborhood for classification and can be easily combined with any neural network. We show that this model outperforms several recently proposed methods for semi-supervised classification in the most thorough study done so far for GCN-like models. Our implementation is available online.},
  langid = {english}
}

@incollection{getoor2005,
  title = {Link-Based {{Classification}}},
  booktitle = {Advanced {{Methods}} for {{Knowledge Discovery}} from {{Complex Data}}},
  author = {Getoor, Lise},
  editor = {Bandyopadhyay, Sanghamitra and Maulik, Ujjwal and Holder, Lawrence B. and Cook, Diane J.},
  year = {2005},
  series = {Advanced {{Information}} and {{Knowledge Processing}}},
  pages = {189--207},
  publisher = {Springer},
  address = {London},
  doi = {10.1007/1-84628-284-5_7},
  urldate = {2024-02-06},
  abstract = {A key challenge for machine learning is the problem of mining richly structured data sets, where the objects are linked in some way due to either an explicit or implicit relationship that exists between the objects. Links among the objects demonstrate certain patterns, which can be helpful for many machine learning tasks and are usually hard to capture with traditional statistical models. Recently there has been a surge of interest in this area, fuelled largely by interest in web and hypertext mining, but also by interest in mining social networks, bibliographic citation data, epidemiological data and other domains best described using a linked or graph structure. In this chapter we propose a framework for modeling link distributions, a link-based model that supports discriminative models describing both the link distributions and the attributes of linked objects. We use a structured logistic regression model, capturing both content and links. We systematically evaluate several variants of our link-based model on a range of data sets including both web and citation collections. In all cases, the use of the link distribution improves classification performance.},
  isbn = {978-1-84628-284-3},
  langid = {english},
  keywords = {Inductive Logic Programming,Link Feature,Link Mining,Remove Stop Word,Unlabeled Data}
}

@inproceedings{giles1998,
  title = {{{CiteSeer}}: An Automatic Citation Indexing System},
  shorttitle = {{{CiteSeer}}},
  booktitle = {Proceedings of the Third {{ACM}} Conference on {{Digital}} Libraries},
  author = {Giles, C. Lee and Bollacker, Kurt D. and Lawrence, Steve},
  year = {1998},
  month = may,
  series = {{{DL}} '98},
  pages = {89--98},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  doi = {10.1145/276675.276685},
  urldate = {2024-02-06},
  isbn = {978-0-89791-965-4}
}

@inproceedings{hu2020,
  title = {Open {{Graph Benchmark}}: {{Datasets}} for {{Machine Learning}} on {{Graphs}}},
  shorttitle = {Open {{Graph Benchmark}}},
  booktitle = {Advances in {{Neural Information Processing Systems}}},
  author = {Hu, Weihua and Fey, Matthias and Zitnik, Marinka and Dong, Yuxiao and Ren, Hongyu and Liu, Bowen and Catasta, Michele and Leskovec, Jure},
  year = {2020},
  volume = {33},
  pages = {22118--22133},
  publisher = {Curran Associates, Inc.},
  urldate = {2023-09-14},
  abstract = {We present the Open Graph Benchmark (OGB), a diverse set of challenging and realistic benchmark datasets to facilitate scalable, robust, and reproducible graph machine learning (ML) research. OGB datasets are large-scale, encompass multiple important graph ML tasks, and cover a diverse range of domains, ranging from social and information networks to biological networks, molecular graphs, source code ASTs, and knowledge graphs. For each dataset, we provide a unified evaluation protocol using meaningful application-specific data splits and evaluation metrics. In addition to building the datasets, we also perform extensive benchmark experiments for each dataset. Our experiments suggest that OGB datasets present significant challenges of scalability to large-scale graphs and out-of-distribution generalization under realistic data splits, indicating fruitful opportunities for future research. Finally, OGB provides an automated end-to-end graph ML pipeline that simplifies and standardizes the process of graph data loading, experimental setup, and model evaluation. OGB will be regularly updated and welcomes inputs from the community. OGB datasets as well as data loaders, evaluation scripts, baseline code, and leaderboards are publicly available at https://ogb.stanford.edu .}
}

@article{huhn2009,
  title = {{{FR3}}: {{A Fuzzy Rule Learner}} for {{Inducing Reliable Classifiers}}},
  shorttitle = {{{FR3}}},
  author = {H{\"u}hn, Jens Christian and H{\"u}llermeier, Eyke},
  year = {2009},
  month = feb,
  journal = {IEEE Transactions on Fuzzy Systems},
  volume = {17},
  number = {1},
  pages = {138--149},
  issn = {1941-0034},
  doi = {10.1109/TFUZZ.2008.2005490},
  urldate = {2024-02-10},
  abstract = {This paper introduces a fuzzy-rule-based classification method called fuzzy round robin [repeated incremental pruning to produce error reduction (RIPPER)] (FR3). As the name suggests, FR3 builds upon the RIPPER algorithm, a state-of-the-art rule learner. More specifically, in the context of polychotomous classification, it uses a fuzzy extension of RIPPER as a base learner within a round robin scheme, and thus, can be seen as a fuzzy variant of the R3 learner that has recently been introduced in the literature. A key feature of FR3, in comparison with its nonfuzzy counterpart, is its ability to represent different facets of uncertainty involved in a classification decision in a more faithful way. FR3 thus provides the basis for implementing ``reliable classifiers'' that may abstain from a decision when not being sure enough, or at least indicate that a classification is not fully supported by the empirical evidence at hand. Besides, our experimental results show that FR3 outperforms R3 in terms of classification accuracy, and therefore, suggest that it produces predictions that are not only more reliable but also more accurate. The superb classification performance of FR3 is furthermore confirmed by comparing it to other state-of-the-art (fuzzy) rule learners.},
  keywords = {Computer science,Decision making,Fuzzy preference structures,fuzzy-rule-based systems,Learning systems,Mathematics,pattern classification,Pattern classification,Probability distribution,Round robin,Uncertainty,uncertainty representation}
}

@article{hullermeier2021,
  title = {Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods},
  shorttitle = {Aleatoric and Epistemic Uncertainty in Machine Learning},
  author = {H{\"u}llermeier, Eyke and Waegeman, Willem},
  year = {2021},
  month = mar,
  journal = {Machine Learning},
  volume = {110},
  number = {3},
  pages = {457--506},
  issn = {1573-0565},
  doi = {10.1007/s10994-021-05946-3},
  urldate = {2023-08-24},
  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.},
  langid = {english},
  keywords = {Bayesian inference,Calibration,Conformal prediction,Credal sets and classifiers,Deep neural networks,Ensembles,Epistemic uncertainty,Gaussian processes,Generative models,Likelihood-based methods,Probability,Set-valued prediction,Uncertainty,Version space learning}
}

@inproceedings{huseljic2021,
  title = {Separation of {{Aleatoric}} and {{Epistemic Uncertainty}} in {{Deterministic Deep Neural Networks}}},
  booktitle = {2020 25th {{International Conference}} on {{Pattern Recognition}} ({{ICPR}})},
  author = {Huseljic, Denis and Sick, Bernhard and Herde, Marek and Kottke, Daniel},
  year = {2021},
  month = jan,
  pages = {9172--9179},
  issn = {1051-4651},
  doi = {10.1109/ICPR48806.2021.9412616},
  urldate = {2024-02-09},
  abstract = {Despite the success of deep neural networks (DNN) in many applications, their ability to model uncertainty is still significantly limited. For example, in safety-critical applications such as autonomous driving, it is crucial to obtain a prediction that reflects different types of uncertainty to address life-threatening situations appropriately. In such cases, it is essential to be aware of the risk (i.e., aleatoric uncertainty) and the reliability (i.e., epistemic uncertainty) that comes with a prediction. We present AE-DNN, a model allowing the separation of aleatoric and epistemic uncertainty while maintaining a proper generalization capability. AE-DNN is based on deterministic DNN, which can determine the respective uncertainty measures in a single forward pass. In analyses with synthetic and image data, we show that our method improves the modeling of epistemic uncertainty while providing an intuitively understandable separation of risk and reliability.},
  keywords = {Analytical models,Data models,Measurement uncertainty,Neural networks,Pattern recognition,Reliability,Uncertainty}
}

@inproceedings{kendall2017,
  title = {What Uncertainties Do We Need in {{Bayesian}} Deep Learning for Computer Vision?},
  booktitle = {Proceedings of the 31st {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Kendall, Alex and Gal, Yarin},
  year = {2017},
  month = dec,
  series = {{{NIPS}}'17},
  pages = {5580--5590},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2024-03-19},
  abstract = {There are two major types of uncertainty one can model. Aleatoric uncertainty captures noise inherent in the observations. On the other hand, epistemic uncertainty accounts for uncertainty in the model - uncertainty which can be explained away given enough data. Traditionally it has been difficult to model epistemic uncertainty in computer vision, but with new Bayesian deep learning tools this is now possible. We study the benefits of modeling epistemic vs. aleatoric uncertainty in Bayesian deep learning models for vision tasks. For this we present a Bayesian deep learning framework combining input-dependent aleatoric uncertainty together with epistemic uncertainty. We study models under the framework with per-pixel semantic segmentation and depth regression tasks. Further, our explicit uncertainty formulation leads to new loss functions for these tasks, which can be interpreted as learned attenuation. This makes the loss more robust to noisy data, also giving new state-of-the-art results on segmentation and depth regression benchmarks.},
  isbn = {978-1-5108-6096-4}
}

@inproceedings{kipf2017,
  title = {Semi-{{Supervised Classification}} with {{Graph Convolutional Networks}}},
  booktitle = {International {{Conference}} on {{Learning Representations}}},
  author = {Kipf, Thomas N. and Welling, Max},
  year = {2017},
  urldate = {2023-08-25},
  abstract = {We present a scalable approach for semi-supervised learning on graph-structured data that is based on an efficient variant of convolutional neural networks which operate directly on graphs. We motivate the choice of our convolutional architecture via a localized first-order approximation of spectral graph convolutions. Our model scales linearly in the number of graph edges and learns hidden layer representations that encode both local graph structure and features of nodes. In a number of experiments on citation networks and on a knowledge graph dataset we demonstrate that our approach outperforms related methods by a significant margin.},
  langid = {english}
}

@article{koliander2022,
  title = {Fusion of {{Probability Density Functions}}},
  author = {Koliander, G{\"u}nther and {El-Laham}, Yousef and Djuri{\'c}, Petar M. and Hlawatsch, Franz},
  year = {2022},
  month = apr,
  journal = {Proceedings of the IEEE},
  volume = {110},
  number = {4},
  pages = {404--453},
  issn = {1558-2256},
  doi = {10.1109/JPROC.2022.3154399},
  urldate = {2024-02-12},
  abstract = {Fusing probabilistic information is a fundamental task in signal and data processing with relevance to many fields of technology and science. In this work, we investigate the fusion of multiple probability density functions (pdfs) of a continuous random variable or vector. Although the case of continuous random variables and the problem of pdf fusion frequently arise in multisensor signal processing, statistical inference, and machine learning, a universally accepted method for pdf fusion does not exist. The diversity of approaches, perspectives, and solutions related to pdf fusion motivates a unified presentation of the theory and methodology of the field. We discuss three different approaches to fusing pdfs. In the axiomatic approach, the fusion rule is defined indirectly by a set of properties (axioms). In the optimization approach, it is the result of minimizing an objective function that involves an information-theoretic divergence or a distance measure. In the supra-Bayesian approach, the fusion center interprets the pdfs to be fused as random observations. Our work is partly a survey, reviewing in a structured and coherent fashion many of the concepts and methods that have been developed in the literature. In addition, we present new results for each of the three approaches. Our original contributions include new fusion rules, axioms, and axiomatic and optimization-based characterizations; a new formulation of supra-Bayesian fusion in terms of finite-dimensional parametrizations; and a study of supra-Bayesian fusion of posterior pdfs for linear Gaussian models.},
  keywords = {-divergence,Bayes methods,Chernoff fusion,covariance intersection,Data processing,forecasting,Holder mean,information fusion,Kullback-Leibler divergence,linear Gaussian model,Linear programming,Machine learning,model averaging,multisensor signal processing,pooling function,Probabilistic logic,probabilistic machine learning,probabilistic opinion pooling,Probability density function,Random variables,sensor network,Signal processing,supra-Bayesian fusion,target tracking.}
}

@inproceedings{kopetzki2021,
  title = {Evaluating {{Robustness}} of {{Predictive Uncertainty Estimation}}: {{Are Dirichlet-based Models Reliable}}?},
  shorttitle = {Evaluating {{Robustness}} of {{Predictive Uncertainty Estimation}}},
  booktitle = {Proceedings of the 38th {{International Conference}} on {{Machine Learning}}},
  author = {Kopetzki, Anna-Kathrin and Charpentier, Bertrand and Z{\"u}gner, Daniel and Giri, Sandhya and G{\"u}nnemann, Stephan},
  year = {2021},
  month = jul,
  pages = {5707--5718},
  publisher = {PMLR},
  issn = {2640-3498},
  urldate = {2024-02-09},
  abstract = {Dirichlet-based uncertainty (DBU) models are a recent and promising class of uncertainty-aware models. DBU models predict the parameters of a Dirichlet distribution to provide fast, high-quality uncertainty estimates alongside with class predictions. In this work, we present the first large-scale, in-depth study of the robustness of DBU models under adversarial attacks. Our results suggest that uncertainty estimates of DBU models are not robust w.r.t. three important tasks: (1) indicating correctly and wrongly classified samples; (2) detecting adversarial examples; and (3) distinguishing between in-distribution (ID) and out-of-distribution (OOD) data. Additionally, we explore the first approaches to make DBU mod- els more robust. While adversarial training has a minor effect, our median smoothing based ap- proach significantly increases robustness of DBU models.},
  langid = {english}
}

@misc{kotelevskii2023,
  title = {Dirichlet-Based {{Uncertainty Quantification}} for {{Personalized Federated Learning}} with {{Improved Posterior Networks}}},
  author = {Kotelevskii, Nikita and Horv{\'a}th, Samuel and Nandakumar, Karthik and Tak{\'a}{\v c}, Martin and Panov, Maxim},
  year = {2023},
  month = dec,
  number = {arXiv:2312.11230},
  eprint = {2312.11230},
  primaryclass = {cs, stat},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2312.11230},
  urldate = {2024-02-08},
  abstract = {In modern federated learning, one of the main challenges is to account for inherent heterogeneity and the diverse nature of data distributions for different clients. This problem is often addressed by introducing personalization of the models towards the data distribution of the particular client. However, a personalized model might be unreliable when applied to the data that is not typical for this client. Eventually, it may perform worse for these data than the non-personalized global model trained in a federated way on the data from all the clients. This paper presents a new approach to federated learning that allows selecting a model from global and personalized ones that would perform better for a particular input point. It is achieved through a careful modeling of predictive uncertainties that helps to detect local and global in- and out-of-distribution data and use this information to select the model that is confident in a prediction. The comprehensive experimental evaluation on the popular real-world image datasets shows the superior performance of the model in the presence of out-of-distribution data while performing on par with state-of-the-art personalized federated learning algorithms in the standard scenarios.},
  archiveprefix = {arxiv},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning}
}

@inproceedings{malinin2018,
  title = {Predictive Uncertainty Estimation via Prior Networks},
  booktitle = {Proceedings of the 32nd {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Malinin, Andrey and Gales, Mark},
  year = {2018},
  month = dec,
  series = {{{NIPS}}'18},
  pages = {7047--7058},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2024-02-08},
  abstract = {Estimating how uncertain an AI system is in its predictions is important to improve the safety of such systems. Uncertainty in predictive can result from uncertainty in model parameters, irreducible data uncertainty and uncertainty due to distributional mismatch between the test and training data distributions. Different actions might be taken depending on the source of the uncertainty so it is important to be able to distinguish between them. Recently, baseline tasks and metrics have been defined and several practical methods to estimate uncertainty developed. These methods, however, attempt to model uncertainty due to distributional mismatch either implicitly through model uncertainty or as data uncertainty. This work proposes a new framework for modeling predictive uncertainty called Prior Networks (PNs) which explicitly models distributional uncertainty. PNs do this by parameterizing a prior distribution over predictive distributions. This work focuses on uncertainty for classification and evaluates PNs on the tasks of identifying out-of-distribution (OOD) samples and detecting misclassification on the MNIST and CIFAR-10 datasets, where they are found to outperform previous methods. Experiments on synthetic and MNIST and CIFAR-10 data show that unlike previous non-Bayesian methods PNs are able to distinguish between data and distributional uncertainty.}
}

@misc{malinin2021,
  title = {Uncertainty {{Estimation}} in {{Autoregressive Structured Prediction}}},
  author = {Malinin, Andrey and Gales, Mark},
  year = {2021},
  month = feb,
  number = {arXiv:2002.07650},
  eprint = {2002.07650},
  primaryclass = {cs, stat},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2002.07650},
  urldate = {2024-02-08},
  abstract = {Uncertainty estimation is important for ensuring safety and robustness of AI systems. While most research in the area has focused on un-structured prediction tasks, limited work has investigated general uncertainty estimation approaches for structured prediction. Thus, this work aims to investigate uncertainty estimation for autoregressive structured prediction tasks within a single unified and interpretable probabilistic ensemble-based framework. We consider: uncertainty estimation for sequence data at the token-level and complete sequence-level; interpretations for, and applications of, various measures of uncertainty; and discuss both the theoretical and practical challenges associated with obtaining them. This work also provides baselines for token-level and sequence-level error detection, and sequence-level out-of-domain input detection on the WMT'14 English-French and WMT'17 English-German translation and LibriSpeech speech recognition datasets.},
  archiveprefix = {arxiv},
  keywords = {Computer Science - Artificial Intelligence,Computer Science - Machine Learning,Statistics - Machine Learning}
}

@inproceedings{mcauley2015,
  title = {Image-{{Based Recommendations}} on {{Styles}} and {{Substitutes}}},
  booktitle = {Proceedings of the 38th {{International ACM SIGIR Conference}} on {{Research}} and {{Development}} in {{Information Retrieval}}},
  author = {McAuley, Julian and Targett, Christopher and Shi, Qinfeng and {van den Hengel}, Anton},
  year = {2015},
  month = aug,
  series = {{{SIGIR}} '15},
  pages = {43--52},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  doi = {10.1145/2766462.2767755},
  urldate = {2024-02-06},
  abstract = {Humans inevitably develop a sense of the relationships between objects, some of which are based on their appearance. Some pairs of objects might be seen as being alternatives to each other (such as two pairs of jeans), while others may be seen as being complementary (such as a pair of jeans and a matching shirt). This information guides many of the choices that people make, from buying clothes to their interactions with each other. We seek here to model this human sense of the relationships between objects based on their appearance. Our approach is not based on fine-grained modeling of user annotations but rather on capturing the largest dataset possible and developing a scalable method for uncovering human notions of the visual relationships within. We cast this as a network inference problem defined on graphs of related images, and provide a large-scale dataset for the training and evaluation of the same. The system we develop is capable of recommending which clothes and accessories will go well together (and which will not), amongst a host of other applications.},
  isbn = {978-1-4503-3621-5},
  keywords = {metric learning,recommender systems,visual features}
}

@article{mccallum2000,
  title = {Automating the {{Construction}} of {{Internet Portals}} with {{Machine Learning}}},
  author = {McCallum, Andrew Kachites and Nigam, Kamal and Rennie, Jason and Seymore, Kristie},
  year = {2000},
  month = jul,
  journal = {Information Retrieval},
  volume = {3},
  number = {2},
  pages = {127--163},
  issn = {1573-7659},
  doi = {10.1023/A:1009953814988},
  urldate = {2024-02-06},
  abstract = {Domain-specific internet portals are growing in popularity because they gather content from the Web and organize it for easy access, retrieval and search. For example, www.campsearch.com allows complex queries by age, location, cost and specialty over summer camps. This functionality is not possible with general, Web-wide search engines. Unfortunately these portals are difficult and time-consuming to maintain. This paper advocates the use of machine learning techniques to greatly automate the creation and maintenance of domain-specific Internet portals. We describe new research in reinforcement learning, information extraction and text classification that enables efficient spidering, the identification of informative text segments, and the population of topic hierarchies. Using these techniques, we have built a demonstration system: a portal for computer science research papers. It already contains over 50,000 papers and is publicly available at www.cora.justresearch.com. These techniques are widely applicable to portal creation in other domains.},
  langid = {english},
  keywords = {crawling,expectation-maximization,hidden Markov models,information extraction,naive Bayes,reinforcement learning,spidering,text classification,unlabeled data}
}

@article{melbourne2022,
  title = {The {{Differential Entropy}} of {{Mixtures}}: {{New Bounds}} and {{Applications}}},
  shorttitle = {The {{Differential Entropy}} of {{Mixtures}}},
  author = {Melbourne, James and Talukdar, Saurav and Bhaban, Shreyas and Madiman, Mokshay and Salapaka, Murti V.},
  year = {2022},
  month = apr,
  journal = {IEEE Transactions on Information Theory},
  volume = {68},
  number = {4},
  pages = {2123--2146},
  issn = {1557-9654},
  doi = {10.1109/TIT.2022.3140661},
  urldate = {2024-01-05},
  abstract = {Mixture distributions are extensively used as a modeling tool in diverse areas from machine learning to communications engineering to physics, and obtaining bounds on the entropy of mixture distributions is of fundamental importance in many of these applications. This article provides sharp bounds on the entropy concavity deficit, which is the difference between the differential entropy of the mixture and the weighted sum of differential entropies of constituent components. Toward establishing lower and upper bounds on the concavity deficit, results that are of importance in their own right are obtained. In order to obtain nontrivial upper bounds, properties of the skew-divergence are developed and notions of ``skew'' f -divergences are introduced; a reverse Pinsker inequality and a bound on Jensen-Shannon divergence are obtained along the way. Complementary lower bounds are derived with special attention paid to the case that corresponds to independent summation of a continuous and a discrete random variable. Several applications of the bounds are delineated, including to mutual information of additive noise channels, thermodynamics of computation, and functional inequalities.}
}

@inproceedings{namata2012,
  title = {Query-Driven {{Active Surveying}} for {{Collective Classification}}},
  author = {Namata, Galileo and London, Ben and Getoor, L. and Huang, Bert},
  year = {2012},
  urldate = {2024-02-06},
  abstract = {In network classification problems such as those found in intelligence gathering, public health, and viral marketing, one is often only interested in inferring the labels of a subset of the nodes. We refer to this subset as the query set, and define the problem as query-driven collective classification. We study this problem in a practical active learning framework, in which the learning algorithm can survey non-query nodes to obtain their labels and network structure. We derive a surveying strategy aimed toward optimal inference on the query set. Considering both feature and structural smoothness, concepts that we formally define, we develop an algorithm which adaptively selects survey nodes by estimating which form of smoothness is most appropriate. We evaluate our algorithm on several network datasets and demonstrate its improvements over standard active learning methods.}
}

@article{pal1993,
  title = {Uncertainty Measures for Evidential Reasoning {{II}}: {{A}} New Measure of Total Uncertainty},
  shorttitle = {Uncertainty Measures for Evidential Reasoning {{II}}},
  author = {Pal, Nikhil R. and Bezdek, James C. and Hemasinha, Rohan},
  year = {1993},
  month = jan,
  journal = {International Journal of Approximate Reasoning},
  volume = {8},
  number = {1},
  pages = {1--16},
  issn = {0888-613X},
  doi = {10.1016/S0888-613X(05)80003-9},
  urldate = {2024-02-08},
  abstract = {In Part I we discussed limitations of two measures of global (non-fuzzy) uncertainty of Lamata and Moral, and a measure of total (non-fuzzy) uncertainty due to Klir and Ramer and established the need for a new measure. In this paper we propose a set of intuitively desirable axioms for a measure of total uncertainty (TU) associated with a basic assignment m(A), and then derive an expression for a (unique) function that satisfies these requirements. Several theorems are proved about the new measure. Our measure is additive, and unlike other TU measures, has a unique maximum. The new measure reduces to Shannon's probabilistic entropy when the basic probability assignment focuses only on singletons. On the other hand, complete ignorance---basic assignment focusing only on the entire set, as a whole---reduces it to Hartley's measure of information. We show that the computational complexity of the new measure is O(N), whereas previous measures of TU are O(N2). Finally, we compare the new measure to its predecessors by extending the numerical example of Part I so that it includes values of the new measure.},
  keywords = {Conflict,confusion,dissonance,entropy,evidential reasoning,specificity,uncertainty}
}

@article{redner1998,
  title = {How Popular Is Your Paper? {{An}} Empirical Study of the Citation Distribution},
  shorttitle = {How Popular Is Your Paper?},
  author = {Redner, S.},
  year = {1998},
  month = jul,
  journal = {The European Physical Journal B - Condensed Matter and Complex Systems},
  volume = {4},
  number = {2},
  pages = {131--134},
  issn = {1434-6036},
  doi = {10.1007/s100510050359},
  urldate = {2024-02-10},
  abstract = {Numerical data for the distribution of citations are examined for: (i) papers published in 1981 in journals which are catalogued by the Institute for Scientific Information (783,339 papers) and (ii) 20 years of publications in Physical Review D, vols. 11-50 (24,296 papers). A Zipf plot of the number of citations to a given paper versus its citation rank appears to be consistent with a power-law dependence for leading rank papers, with exponent close to -1/2. This, in turn, suggests that the number of papers with x citations, N(x), has a large-x power law decay \$\$N(x) {\textbackslash}sim \{x\^{}\{ - a\}\}\$\$, with \$\$a {\textbackslash}approx 3\$\$.},
  langid = {english},
  keywords = {PACS. 02.50.-r Probability theory stochastic processes and statistics - 01.75.+m Science and society - 89.90.+n Other areas of general interest to physicists}
}

@misc{sale2023a,
  title = {Second-{{Order Uncertainty Quantification}}: {{Variance-Based Measures}}},
  shorttitle = {Second-{{Order Uncertainty Quantification}}},
  author = {Sale, Yusuf and Hofman, Paul and Wimmer, Lisa and H{\"u}llermeier, Eyke and Nagler, Thomas},
  year = {2023},
  month = dec,
  number = {arXiv:2401.00276},
  eprint = {2401.00276},
  primaryclass = {cs, stat},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2401.00276},
  urldate = {2024-01-17},
  abstract = {Uncertainty quantification is a critical aspect of machine learning models, providing important insights into the reliability of predictions and aiding the decision-making process in real-world applications. This paper proposes a novel way to use variance-based measures to quantify uncertainty on the basis of second-order distributions in classification problems. A distinctive feature of the measures is the ability to reason about uncertainties on a class-based level, which is useful in situations where nuanced decision-making is required. Recalling some properties from the literature, we highlight that the variance-based measures satisfy important (axiomatic) properties. In addition to this axiomatic approach, we present empirical results showing the measures to be effective and competitive to commonly used entropy-based measures.},
  archiveprefix = {arxiv},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning}
}

@inproceedings{schweighofer2023,
  title = {Introducing an {{Improved Information-Theoretic Measure}} of {{Predictive Uncertainty}}},
  booktitle = {{{NeurIPS}} 2023 {{Workshop}} on {{Mathematics}} of {{Modern Machine Learning}}},
  author = {Schweighofer, Kajetan and Aichberger, Lukas and Ielanskyi, Mykyta and Hochreiter, Sepp},
  year = {2023},
  month = nov,
  urldate = {2024-02-09},
  abstract = {Applying a machine learning model for decision-making in the real world requires to distinguish what the model knows from what it does not. A critical factor in assessing the knowledge of a model is to quantify its predictive uncertainty. Predictive uncertainty is commonly measured by the entropy of the Bayesian model average (BMA) predictive distribution. Yet, the properness of this current measure of predictive uncertainty was recently questioned. We provide new insights regarding those limitations. Our analyses show that the current measure erroneously assumes that the BMA predictive distribution is equivalent to the predictive distribution of the true model that generated the dataset. Consequently, we introduce a theoretically grounded measure to overcome these limitations. We experimentally verify the benefits of our introduced measure of predictive uncertainty. We find that our introduced measure behaves more reasonably in controlled synthetic tasks. Moreover, our evaluations on ImageNet demonstrate that our introduced measure is advantageous in real-world applications utilizing predictive uncertainty.},
  langid = {english}
}

@article{sen2008,
  title = {Collective {{Classification}} in {{Network Data}}},
  author = {Sen, Prithviraj and Namata, Galileo and Bilgic, Mustafa and Getoor, Lise and Galligher, Brian and {Eliassi-Rad}, Tina},
  year = {2008},
  month = sep,
  journal = {AI Magazine},
  volume = {29},
  number = {3},
  pages = {93--93},
  issn = {2371-9621},
  doi = {10.1609/aimag.v29i3.2157},
  urldate = {2024-02-06},
  abstract = {Many real-world applications produce networked data such as the world-wide web (hypertext documents connected via hyperlinks), social networks (for example, people connected by friendship links), communication networks (computers connected via communication links) and biological networks (for example, protein interaction networks). A recent focus in machine learning research has been to extend traditional machine learning classification techniques to classify nodes in such networks. In this article, we provide a brief introduction to this area of research and how it has progressed during the past decade. We introduce four of the most widely used inference algorithms for classifying networked data and empirically compare them on both synthetic and real-world data.},
  copyright = {Copyright (c) 0},
  langid = {english}
}

@inproceedings{sensoy2018,
  title = {Evidential Deep Learning to Quantify Classification Uncertainty},
  booktitle = {Proceedings of the 32nd {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Sensoy, Murat and Kaplan, Lance and Kandemir, Melih},
  year = {2018},
  month = dec,
  series = {{{NIPS}}'18},
  pages = {3183--3193},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2024-02-08},
  abstract = {Deterministic neural nets have been shown to learn effective predictors on a wide range of machine learning problems. However, as the standard approach is to train the network to minimize a prediction loss, the resultant model remains ignorant to its prediction confidence. Orthogonally to Bayesian neural nets that indirectly infer prediction uncertainty through weight uncertainties, we propose explicit modeling of the same using the theory of subjective logic. By placing a Dirichlet distribution on the class probabilities, we treat predictions of a neural net as subjective opinions and learn the function that collects the evidence leading to these opinions by a deterministic neural net from data. The resultant predictor for a multi-class classification problem is another Dirichlet distribution whose parameters are set by the continuous output of a neural net. We provide a preliminary analysis on how the peculiarities of our new loss function drive improved uncertainty estimation. We observe that our method achieves unprecedented success on detection of out-of-distribution queries and endurance against adversarial perturbations.}
}

@article{shu2017,
  title = {Fake {{News Detection}} on {{Social Media}}: {{A Data Mining Perspective}}},
  shorttitle = {Fake {{News Detection}} on {{Social Media}}},
  author = {Shu, Kai and Sliva, Amy and Wang, Suhang and Tang, Jiliang and Liu, Huan},
  year = {2017},
  month = sep,
  journal = {ACM SIGKDD Explorations Newsletter},
  volume = {19},
  number = {1},
  pages = {22--36},
  issn = {1931-0145},
  doi = {10.1145/3137597.3137600},
  urldate = {2024-02-06},
  abstract = {Social media for news consumption is a double-edged sword. On the one hand, its low cost, easy access, and rapid dissemination of information lead people to seek out and consume news from social media. On the other hand, it enables the wide spread of {\textbackslash}fake news", i.e., low quality news with intentionally false information. The extensive spread of fake news has the potential for extremely negative impacts on individuals and society. Therefore, fake news detection on social media has recently become an emerging research that is attracting tremendous attention. Fake news detection on social media presents unique characteristics and challenges that make existing detection algorithms from traditional news media ine ective or not applicable. First, fake news is intentionally written to mislead readers to believe false information, which makes it difficult and nontrivial to detect based on news content; therefore, we need to include auxiliary information, such as user social engagements on social media, to help make a determination. Second, exploiting this auxiliary information is challenging in and of itself as users' social engagements with fake news produce data that is big, incomplete, unstructured, and noisy. Because the issue of fake news detection on social media is both challenging and relevant, we conducted this survey to further facilitate research on the problem. In this survey, we present a comprehensive review of detecting fake news on social media, including fake news characterizations on psychology and social theories, existing algorithms from a data mining perspective, evaluation metrics and representative datasets. We also discuss related research areas, open problems, and future research directions for fake news detection on social media.}
}

@misc{stadler2021,
  title = {Graph {{Posterior Network}}: {{Bayesian Predictive Uncertainty}} for {{Node Classification}}},
  shorttitle = {Graph {{Posterior Network}}},
  author = {Stadler, Maximilian and Charpentier, Bertrand and Geisler, Simon and Z{\"u}gner, Daniel and G{\"u}nnemann, Stephan},
  year = {2021},
  month = oct,
  number = {arXiv:2110.14012},
  eprint = {2110.14012},
  primaryclass = {cs, stat},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2110.14012},
  urldate = {2023-08-25},
  abstract = {The interdependence between nodes in graphs is key to improve class predictions on nodes and utilized in approaches like Label Propagation (LP) or in Graph Neural Networks (GNN). Nonetheless, uncertainty estimation for non-independent node-level predictions is under-explored. In this work, we explore uncertainty quantification for node classification in three ways: (1) We derive three axioms explicitly characterizing the expected predictive uncertainty behavior in homophilic attributed graphs. (2) We propose a new model Graph Posterior Network (GPN) which explicitly performs Bayesian posterior updates for predictions on interdependent nodes. GPN provably obeys the proposed axioms. (3) We extensively evaluate GPN and a strong set of baselines on semi-supervised node classification including detection of anomalous features, and detection of left-out classes. GPN outperforms existing approaches for uncertainty estimation in the experiments.},
  archiveprefix = {arxiv},
  keywords = {Computer Science - Machine Learning,Statistics - Machine Learning}
}

@article{stone1961,
  title = {The {{Opinion Pool}}},
  author = {Stone, M.},
  year = {1961},
  journal = {The Annals of Mathematical Statistics},
  volume = {32},
  number = {4},
  eprint = {2237933},
  eprinttype = {jstor},
  pages = {1339--1342},
  publisher = {Institute of Mathematical Statistics},
  issn = {0003-4851},
  urldate = {2024-02-10},
  abstract = {When a group of k individuals is required to make a joint decision, it occasionally happens that there is agreement on a utility function for the problem but that opinions differ on the probabilities of the relevant states of nature. When the latter are indexed by a parameter {\texttheta}, to which probability density functions on some measure {$\mu$}({\texttheta}) may be attributed, suppose the k opinions are given by probability density functions ps1({\texttheta}), {$\cdots$}, psk({\texttheta}). Suppose that D is the set of available decisions d and that the utility of d, when the state of nature is {\texttheta}, is u(d, {\texttheta}). For a probability density function p({\texttheta}), write u[ d{$\mid$} p({\texttheta})] = {$\int$} u(d, {\texttheta})p({\texttheta}) d{$\mu$}({\texttheta}). The Group Minimax Rule of Savage [1] would have the group select that d minimising maxi = 1, {$\cdots$}, k\{maxd'{$\varepsilon$} D u[ d' {$\mid$} psi({\texttheta})] - u[ d {$\mid$} psi({\texttheta})]\}. As Savage remarks ([1], p. 175), this rule is undemocratic in that it depends only on the different distributions for {\texttheta} represented in those put forward by the group and not on the number of members of the group supporting each different representative. An alternative rule for choosing d may be stated as follows: "Choose weights {$\lambda$}1, {$\cdots$}, {$\lambda$}k ({$\lambda$}i {$\geq$} 0, i = 1, {$\cdots$}, k and {$\sum$}k 1 {$\lambda$}i = 1); construct the pooled density function ps{$\lambda$}({\texttheta}) = {$\sum$}k 1 {$\lambda$}ip si({\texttheta}); choose the d, say ds{$\lambda$}, maximising u[ d {$\mid$} ps{$\lambda$}({\texttheta})]." This rule, which may be called the Opinion Pool, can be made democratic by setting {$\lambda$}1 = {$\cdots$} = {$\lambda$}k = 1/k. Where it is reasonable to suppose that there is an actual, operative probability distribution, represented by an `unknown' density function pa({\texttheta}), it is clear that the group is then acting as if pa({\texttheta}) were known to be ps{$\lambda$}({\texttheta}). If pa({\texttheta}) were known, it would be possible to calculate u[ ds{$\lambda$} {$\mid$} pa({\texttheta})] and u[ dsi {$\mid$} pa({\texttheta})], where dsi is the d maximising u[ d {$\mid$} psi({\texttheta})], i = 1, {$\cdots$}, k and then to use these quantities to assess the effect of adopting the Opinion Pool for any given choice of {$\lambda$}1, {$\cdots$}, {$\lambda$}k. It is of general theoretical interest to examine the conditions under which {\textbackslash}begin\{equation*\}{\textbackslash}tag\{1.1\}u{\textbackslash}lbrack d\_\{s{\textbackslash}lambda\} {\textbackslash}mid p\_a({\textbackslash}theta){\textbackslash}rbrack {\textbackslash}geqq {\textbackslash}min\_\{i = 1, {\textbackslash}cdots, k\} u{\textbackslash}lbrack d\_\{si\} {\textbackslash}mid p\_a({\textbackslash}theta){\textbackslash}rbrack.{\textbackslash}end\{equation*\} Theorems 2.1 and 3.1 provide different sets of sufficient conditions for (1.1) to hold. Theorem 2.1 requires k = 2 and places a restriction on pa({\texttheta}) (or, equivalently, on ps1({\texttheta}) and ps2({\texttheta})); Theorem 3.1 puts conditions on D and u(d, {\texttheta}) instead.}
}

@article{wang2008,
  title = {Measuring the Preferential Attachment Mechanism in Citation Networks},
  author = {Wang, Mingyang and Yu, Guang and Yu, Daren},
  year = {2008},
  month = jul,
  journal = {Physica A: Statistical Mechanics and its Applications},
  volume = {387},
  number = {18},
  pages = {4692--4698},
  issn = {0378-4371},
  doi = {10.1016/j.physa.2008.03.017},
  urldate = {2024-02-10},
  abstract = {In this paper, we investigated the preferential attachment mechanism (PAM) by considering the dynamic property in papers' in-degree k for three citation networks. We found that the past citations obtained in different years will have different influences on papers' attachment rate {$\Pi$}(k,t). We proposed two methods to consider these different influences. One is the Gradually-vanishing Memory Preferential Attachment Mechanism (GMPAM) based on weighted past citations. The other is the Short-term Memory Preferential Attachment Mechanism (SMPAM) based on citations obtained in the recent one-year period. Experiments showed that SMPAM is simpler and more universal in practice. We can just calculate the citations to papers in the recent one-year period to study the papers' attachment property.},
  keywords = {Citation network,Preferential attachment mechanism,Time window}
}

@inproceedings{wimmer2023,
  title = {Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: {{Are}} Conditional Entropy and Mutual Information Appropriate Measures?},
  shorttitle = {Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning},
  booktitle = {Proceedings of the {{Thirty-Ninth Conference}} on {{Uncertainty}} in {{Artificial Intelligence}}},
  author = {Wimmer, Lisa and Sale, Yusuf and Hofman, Paul and Bischl, Bernd and H{\"u}llermeier, Eyke},
  year = {2023},
  month = jul,
  pages = {2282--2292},
  publisher = {PMLR},
  issn = {2640-3498},
  urldate = {2023-08-31},
  abstract = {The quantification of aleatoric and epistemic uncertainty in terms of conditional entropy and mutual information, respectively, has recently become quite common in machine learning. While the properties of these measures, which are rooted in information theory, seem appealing at first glance, we identify various incoherencies that call their appropriateness into question. In addition to the measures themselves, we critically discuss the idea of an additive decomposition of total uncertainty into its aleatoric and epistemic constituents. Experiments across different computer vision tasks support our theoretical findings and raise concerns about current practice in uncertainty quantification.},
  langid = {english}
}

@inproceedings{xu2018,
  title = {How {{Powerful}} Are {{Graph Neural Networks}}?},
  booktitle = {International {{Conference}} on {{Learning Representations}}},
  author = {Xu, Keyulu and Hu, Weihua and Leskovec, Jure and Jegelka, Stefanie},
  year = {2018},
  month = sep,
  urldate = {2023-08-25},
  abstract = {Graph Neural Networks (GNNs) are an effective framework for representation learning of graphs. GNNs follow a neighborhood aggregation scheme, where the representation vector of a node is computed by recursively aggregating and transforming representation vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs to capture different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.},
  langid = {english}
}

@inproceedings{zhao2020,
  title = {Uncertainty Aware Semi-Supervised Learning on Graph Data},
  booktitle = {Proceedings of the 34th {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Zhao, Xujiang and Chen, Feng and Hu, Shu and Cho, Jin-Hee},
  year = {2020},
  month = dec,
  series = {{{NIPS}}'20},
  pages = {12827--12836},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2023-08-31},
  abstract = {Thanks to graph neural networks (GNNs), semi-supervised node classification has shown the state-of-the-art performance in graph data. However, GNNs have not considered different types of uncertainties associated with class probabilities to minimize risk of increasing misclassification under uncertainty in real life. In this work, we propose a multi-source uncertainty framework using a GNN that reflects various types of predictive uncertainties in both deep learning and belief/evidence theory domains for node classification predictions. By collecting evidence from the given labels of training nodes, the Graph-based Kernel Dirichlet distribution Estimation (GKDE) method is designed for accurately predicting node-level Dirichlet distributions and detecting out-of-distribution (OOD) nodes. We validated the outperformance of our proposed model compared to the state-of-the-art counterparts in terms of misclassification detection and OOD detection based on six real network datasets. We found that dissonance-based detection yielded the best results on misclassification detection while vacuity-based detection was the best for OOD detection. To clarify the reasons behind the results, we provided the theoretical proof that explains the relationships between different types of uncertainties considered in this work.},
  isbn = {978-1-71382-954-6}
}

@inproceedings{zhu2020,
  title = {Beyond Homophily in Graph Neural Networks: Current Limitations and Effective Designs},
  shorttitle = {Beyond Homophily in Graph Neural Networks},
  booktitle = {Proceedings of the 34th {{International Conference}} on {{Neural Information Processing Systems}}},
  author = {Zhu, Jiong and Yan, Yujun and Zhao, Lingxiao and Heimann, Mark and Akoglu, Leman and Koutra, Danai},
  year = {2020},
  month = dec,
  series = {{{NIPS}}'20},
  pages = {7793--7804},
  publisher = {Curran Associates Inc.},
  address = {Red Hook, NY, USA},
  urldate = {2024-02-06},
  abstract = {We investigate the representation power of graph neural networks in the semi-supervised node classification task under heterophily or low homophily, i.e., in networks where connected nodes may have different class labels and dissimilar features. Many popular GNNs fail to generalize to this setting, and are even outperformed by models that ignore the graph structure (e.g., multilayer percep-trons). Motivated by this limitation, we identify a set of key designs---ego- and neighbor-embedding separation, higher-order neighborhoods, and combination of intermediate representations---that boost learning from the graph structure under heterophily. We combine them into a graph neural network, H2GCN, which we use as the base method to empirically evaluate the effectiveness of the identified designs. Going beyond the traditional benchmarks with strong homophily, our empirical analysis shows that the identified designs increase the accuracy of GNNs by up to 40\% and 27\% over models without them on synthetic and real networks with heterophily, respectively, and yield competitive performance under homophily.},
  isbn = {978-1-71382-954-6}
}
