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Distribution-based and credal-set uncertainty representations can be compared fairly when built from the same predictive distributions, and under like-for-like comparison neither paradigm dominates: the winner depends on the measure and tas

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2026-08-02 20:37 UTC pith:Z3SMTBGF

load-bearing objection A genuinely useful controlled comparison of distribution-based vs credal uncertainty representations whose main qualitative finding holds up, but the credal-set construction is confounded with ensemble size M across model families, so the aggregate rankings need qualification. the 2 major comments →

arxiv 2602.22747 v2 pith:Z3SMTBGF submitted 2026-02-26 cs.LG

Set-based v.s. Distribution-based Representations of Epistemic Uncertainty: A Comparative Study

Kaizheng Wang , Yunjia Wang , Fabio Cuzzolin , David Moens , Hans Hallez , Siu Lun Chau This is my paper
classification cs.LG MSC 68T3768T07
keywords epistemic uncertaintycredal setsBayesian neural networksdeep ensemblesuncertainty measuresselective predictionout-of-distribution detectionuncertainty quantification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Two ways of representing a neural network's epistemic uncertainty—a distribution over possible predictive distributions, and a credal set (a convex set of distributions)—are usually treated as incomparable. This paper shows they can be compared like-for-like when both are built from the same finite set of predictive distributions produced by one shared network. Under that protocol, across six uncertainty measures and multiple benchmarks, neither representation wins outright: the Wasserstein distance (distribution-based) leads on selective prediction, the Generalized Hartley measure (credal) leads on out-of-distribution detection, and the entropy-difference measure Hdiff trails in most settings. The authors' conclusion is that claims about uncertainty representations only mean something when the uncertainty measure, benchmark, model, and dataset are specified.

Core claim

Stated in the paper's own terms, the central finding is that 'no representation exhibits uniform superiority independent of the associated uncertainty measure; conclusions depend critically on the interaction between representation and metric.' The authors establish this by deriving both the distribution-based representation (a uniform distribution over M predictive distributions) and the credal representation (the class-wise probability-interval hull of the same M distributions) from an identical set of predictions generated by a shared neural network, eliminating confounding from model differences. Using pairwise signed-rank tests over 10 runs, they find that the distribution-based Wassers

What carries the argument

The core device is a shared basis: a finite collection of predictive distributions B = {p_m}_{m=1}^M obtained from M stochastic forward passes or ensemble members of one neural network. The distribution-based representation is the uniform distribution over B; the credal representation is the set of all probability vectors whose k-th class probability lies between the minimum and maximum of the k-th class probabilities across B. Because both representations come from the same predictions, any performance difference is attributable to the representation or the measure, not the underlying model. Each representation is then scored by three measures—MI, LWV, and WD for the distribution; Hdiff, GH

Load-bearing premise

The credal set is built as the per-class minimum–maximum probability hull of a small sample of M predictive distributions; if those sample extrema misrepresent the model's true epistemic uncertainty, the reported differences between credal and distribution-based representations may be artifacts of interval estimation rather than of the set-based paradigm.

What would settle it

Run the same benchmarks with each predictive model using the same number of predictive samples M (e.g., M = 10), and also vary M for a fixed ensemble (e.g., 2, 5, 10, 20). If the net-win ordering between Wasserstein distance and Generalized Hartley changes with M, or if credal measures' OOD AUROC for identical test points shifts systematically with M, the small-sample min/max hull is a confound and the reported representational rankings do not hold.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Claims that one uncertainty representation is better than another must be qualified by the uncertainty measure, benchmark, predictive model, and dataset; unqualified claims are unsupported by this evidence.
  • Out-of-distribution detection is the more sensitive test for comparing representations; selective-prediction scores are too compressed to expose differences.
  • Within each paradigm there is a clear ordering: Wasserstein distance dominates the other distribution-based measures, and the Generalized Hartley measure dominates the other credal measures, while Hdiff underperforms in most settings.
  • Reliable uncertainty quantification requires both a well-designed representation and a well-designed measure; improving one alone is not enough.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The credal set is formed from the min/max of only M sampled distributions (M = 4, 5, or 10 depending on model), so the interval width — and every credal measure — depends on M. A re-run with all models using the same M would test whether some cross-model differences are artifacts of sample size rather than of the set-based paradigm.
  • If the findings generalize, the 'distribution vs set' debate in epistemic-uncertainty research is partly mis-posed: the dominant factor is which numerical measure is computed, not which representation supplies it. This suggests future work should focus on designing measures that directly target the behavior a task requires.
  • The shared-basis protocol could be applied to other credal constructions, such as likelihood-based or relative-likelihood credal sets, to check whether the Generalized Hartley measure's OOD-detection lead is a property of interval-hull credal sets specifically or of set-based representations generally.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper presents a controlled empirical comparison of two second-order uncertainty representations — a distribution-based representation (a uniform distribution over a finite set of predictive distributions) and a set-based, credal representation (a class-wise probability-interval hull of the same predictive distributions). Both are derived from the same M stochastic forward passes of a shared neural network, which is intended to isolate representational effects from predictive accuracy. Six uncertainty measures are evaluated (MI, LWV, WD for the distribution-based side; Hdiff, GH, MMI for the credal side) across six predictive model families, four selective-prediction settings, and four OOD-detection settings, with 10 independent runs per configuration. The main findings are that no representation dominates uniformly; performance depends on the measure-task interaction (WD leading distribution-based measures on selective prediction, GH leading credal measures on OOD detection, Hdiff performing worst in most settings); and OOD detection exposes representational differences more sharply than selective prediction.

Significance. If the empirical claims hold, this is a useful contribution to the uncertainty-quantification literature. The controlled protocol — building both representations from identical predictive samples — is an important step toward apples-to-apples comparisons of paradigms that are frequently treated as non-comparable. The study is broad (6 model families, 8 task/dataset settings, multiple measures) and the results are reported transparently with per-model tables, statistical significance plots, and reproducibility-oriented implementation details. The observation that rankings are conditional on the measure/task combination is a valuable caution for practitioners. The main threat to the central claim is the uncontrolled variation in M (the number of predictive distributions) across model families, which may artifactually influence the credal measures.

major comments (2)
  1. [§3, §2.2, Eqs. (4)–(5)] The aggregate net-win comparisons in Tables 1–2 and A.4–A.7 pool across model families with different M (DE uses M=5; BatchEns/MaskEns/PackEns use M=4; SVI/MCDO use M=10). Because the credal set is the coordinate-wise min/max hull of the M predictive distributions, adding a sample can only decrease pL_k and increase pU_k; the interval — and therefore Hdiff, GH, and MMI — is a monotone function of M. Thus, the credal measures are not directly comparable across the M-heterogeneous rows that are aggregated in the headline net-win tables. The like-for-like claim requires either holding M fixed across all model families or providing a sensitivity analysis (e.g., recomputing the key comparisons for a representative model at M=4, 5, 10, or subsampling all models to a common M). Without this, the reported GH-over-WD and Hdiff-underperformance patterns could be partly an artifact of the interval-
  2. [§2.2, Eqs. (4)–(5)] The construction of the credal set from a small finite sample (M=4–10) is used without any justification that the min/max hull is a reliable estimator of the true predictive support. For small M, the sample min/max is a biased estimate of the support, systematically understating interval width and hence credal uncertainty. The paper cites De Campos et al. for the interval-credal formalism, but that reference addresses interval probabilities in general, not the small-sample behavior of this specific empirical construction. Since every credal measure is computed from this hull, the paper should either provide evidence that the qualitative rankings are insensitive to M or explicitly frame the conclusions as conditional on the chosen estimator and discuss its potential bias. This is load-bearing for the central 'like-for-like comparison' claim.
minor comments (6)
  1. [§2.2, Eq. (5)] The definition of the credal set omits the probability-simplex constraint. As written, K contains vectors whose components may sum to a value different from 1. The optimization in Eq. (11) later adds the sum-to-1 constraint, but the formal definition should include p ∈ Δ^{K−1} (or the sum constraint) to be mathematically correct.
  2. [§2.2, Eq. (4)] The notation is inconsistent: pn,k is used in the lower-bound expression while pk,m is used in the upper bound. Please unify to pk,m or define n = m.
  3. [§4.1] The text says 'Label-wise Variance (LWV) in (8)' and 'Wasserstein Distance (WD) in (9)', but LWV is given in Eq. (7) and WD in Eq. (8). The equation numbers should be corrected.
  4. [§2.3] Typo: 'miximize' (should be 'minimize') after Eq. (11); 'meidian' (should be 'median') in Eq. (9).
  5. [Tables 1–2] The caption should clarify that the 'Average scores' are means of the six model-specific means, and the reported standard deviations are across those six model means (as opposed to pooled per-run standard deviations). This is inferable from the appendix but should be stated in the main text.
  6. [§4.1] The pairwise one-sided Wilcoxon tests are performed at the 5% level without any correction for multiple comparisons (15 pairwise tests per model per task). The large number of tests may inflate the number of 'significant' wins. A simple multiple-testing correction or an explicit statement that the net-win summaries are exploratory would strengthen the analysis.

Circularity Check

0 steps flagged

No significant circularity: the paper is an empirical comparison; its results come from fresh evaluations, and the self-citations supply building-block formulas but no load-bearing identification.

full rationale

The paper's central claims are empirical findings obtained from fresh runs on held-out tasks, not quantities derived from its own outputs. Both representations are intentionally constructed from the same finite set of predictive distributions (Eqs. 3-5): B is the shared input, the distribution-based measures (Eqs. 6-8) are explicit functions of B, and the credal measures (Eqs. 10-16) are explicit functions of K, which itself is defined from B. No reported prediction is equal by construction to a fitted parameter or to a prior result. The self-citations (e.g., Wang et al. 2024a for the GH algorithm, Chau et al. 2025a for MMI, Wang et al. 2026a for conceptual framing) are not load-bearing: the definitions and algorithms are reproduced in the paper, and the tables report new aggregate Wilcoxon tests over 10 independent runs. The Limitations section transparently restricts the study to probability-interval credal sets induced by finite predictive samples, acknowledging that this is one construction among several; this is a scope choice, not a circular identification. The M-dependence of the min/max hull (Eqs. 4-5 with M=4,5,10 across model families) is a real methodological caveat because the credal interval width is a monotone function of M, but it is a validity/confounding concern rather than a definitional reduction. Moreover, the per-family tables (A.4-A.7) show the qualitative patterns (GH ahead of WD on OOD; Hdiff usually worst) within each individual M value, so the aggregate results are not forced solely by ensemble size. Accordingly, no circular step can be exhibited under the stated standards.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest on four kinds of premises: (i) the BMA approximation — that M forward passes faithfully represent epistemic uncertainty; (ii) the credal construction — that min/max probability intervals over M ≤ 10 samples form a valid credal set, with no analysis of small-sample bias; (iii) the evaluation proxy — that AUARC/AUROC on the chosen tasks measure EU quality even though ground-truth EU is unobservable; and (iv) the statistical protocol — 10 paired runs with one-sided Wilcoxon tests. Free parameters are design choices (M; bounding-box expansion ratios) rather than fitted values. No invented entities. The paper is explicit about scope (Limitations section) but does not quantify the M-dependence of the interval representations.

free parameters (3)
  • Sample/ensemble size M = 4 (BatchEns/MaskEns/PackEns), 5 (DE), 10 (SVI/MCDO)
    Hand-chosen per model family; the credal interval width in Eq. (4) scales with M, so comparisons across models are partially confounded with M.
  • Bounding-box expansion ratios α_min, α_max = 0.25, 0.50
    Hand-chosen in the SeaShip/SMD/SSAVE crop pipeline (Appendix A.3); determines the composition of the ship classification datasets.
  • Crop area thresholds = 256×256 px (SeaShip), 128×128 px (SMD/SSAVE)
    Hand-chosen filters in the dataset pipeline; affects dataset size and difficulty.
axioms (5)
  • domain assumption M stochastic forward passes from a BNN/ensemble provide a faithful finite approximation of the parameter posterior (BMA).
    Invoked in §2.2 Eqs (1)–(3); underlies the definition of B = {p_m}.
  • domain assumption Per-class min/max probability intervals over M samples define a valid and faithful credal set (De Campos et al., 1994).
    §2.2 Eqs (4)–(5); the small-M behavior of extrema is unanalyzed, and different models use different M (4, 5, 10).
  • domain assumption Ground-truth epistemic uncertainty is unobservable, so selective prediction (AUARC) and OOD detection (AUROC) are valid proxies for EU quality.
    §2.4; also imports the OOD caveat from Li et al. 2025.
  • domain assumption Ten independent runs per configuration suffice for paired one-sided Wilcoxon tests at α=0.05 without multiplicity correction.
    §4.1; 15 pairwise comparisons per model, no adjustment.
  • domain assumption Entropy/log-loss and squared-error decompositions (Eqs 6–7) and the WD/interval measures used are the relevant EU definitions.
    Adopted from Hüllermeier & Waegeman 2021, Sale et al. 2024a,b, Abellán et al. 2006, Chau et al. 2025a; the paper compares these measures but does not justify their completeness as the EU semantic.

pith-pipeline@v1.3.0-alltime-deepseek · 22082 in / 18840 out tokens · 170087 ms · 2026-08-02T20:37:02.495590+00:00 · methodology

0 comments
read the original abstract

Epistemic uncertainty in neural networks is commonly modeled using two second-order paradigms: distribution-based representations, which rely on posterior parameter distributions, and set-based representations based on credal sets (convex sets of probability distributions). These frameworks are often regarded as fundamentally non-comparable due to differing semantics, assumptions, and evaluation practices, leaving their relative merits unclear. Empirical comparisons are further confounded by variations in the underlying predictive models. To clarify this issue, we present a controlled comparative study enabling principled, like-for-like evaluation of the two paradigms. Both representations are constructed from the same finite collection of predictive distributions generated by a shared neural network, isolating representational effects from predictive accuracy. Our study evaluates each representation through the lens of 3 uncertainty measures across 8 benchmarks, including selective prediction and out-of-distribution detection, spanning 6 underlying predictive models and 10 independent runs per configuration. Our results show that meaningful comparison between these seemingly non-comparable frameworks is both feasible and informative, providing insights into how second-order representation choices impact practical uncertainty-aware performance.

Figures

Figures reproduced from arXiv: 2602.22747 by David Moens, Fabio Cuzzolin, Hans Hallez, Kaizheng Wang, Siu Lun Chau, Yunjia Wang.

Figure 1
Figure 1. Figure 1: Illustration of our comparative study framework. Two dominant paradigms have emerged for representing such second-order uncertainty. The first adopts a distribution-based representation, where uncertainty is modeled via probability distributions over model parameters or predictions. This perspective underlies Bayesian neural networks (BNNs) as well as practical approximations such as deep ensembles (DE) [B… view at source ↗
Figure 2
Figure 2. Figure 2: Statistical significance plots on different selective prediction (a, b) and OOD detection (c, d) benchmarks across different underlying predictive models. A cell is shaded if the measure in the i-th row is statistically significantly better than that in the j-th column according to a pairwise one-sided Wilcoxon signed-rank test at the 5% significance level. Intra-representation comparisons are shown in blu… view at source ↗
Figure 3
Figure 3. Figure 3: WSI (patient_017_node_2) with ground-truth annotations and model predictions. Predictions have been generated from a ResNet Ensemble. The tumor confidence decreases in areas near the border of the annotation, while the uncertainty thereby increases. This result is consistent over the whole dataset. outperformed their singular counterparts under the domain shift sce￾narios. Further analysis of the ranking o… view at source ↗

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Reference graph

Works this paper leans on

54 extracted references · 1 canonical work pages

  1. [1]

    Deep ensembles work, but are they necessary? Advances in Neural Information Processing Systems, 35: 0 33646--33660, 2022

    Taiga Abe, Estefany Kelly Buchanan, Geoff Pleiss, Richard Zemel, and John P Cunningham. Deep ensembles work, but are they necessary? Advances in Neural Information Processing Systems, 35: 0 33646--33660, 2022

    2022

  2. [2]

    A non-specificity measure for convex sets of probability distributions

    Joaqu \' n Abell \'a n and Seraf \' n Moral. A non-specificity measure for convex sets of probability distributions. International journal of uncertainty, fuzziness and knowledge-based systems, 8 0 (03): 0 357--367, 2000

    2000

  3. [3]

    Disaggregated total uncertainty measure for credal sets

    Joaqu \' n Abell \'a n, George J Klir, and Seraf \' n Moral. Disaggregated total uncertainty measure for credal sets. International Journal of General Systems, 35 0 (1): 0 29--44, 2006

    2006

  4. [4]

    Benchmarking Bayesian deep learning on diabetic retinopathy detection tasks

    Neil Band, Tim GJ Rudner, Qixuan Feng, Angelos Filos, Zachary Nado, Michael W Dusenberry, Ghassen Jerfel, Dustin Tran, and Yarin Gal. Benchmarking Bayesian deep learning on diabetic retinopathy detection tasks. In Proceedings of Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2), 2021

    2021

  5. [5]

    From detection of individual metastases to classification of lymph node status at the patient level: the camelyon17 challenge

    Peter Bandi, Oscar Geessink, Quirine Manson, Marcory Van Dijk, Maschenka Balkenhol, Meyke Hermsen, Babak Ehteshami Bejnordi, Byungjae Lee, Kyunghyun Paeng, Aoxiao Zhong, et al. From detection of individual metastases to classification of lymph node status at the patient level: the camelyon17 challenge. IEEE transactions on medical imaging, 38 0 (2): 0 550...

    2018

  6. [6]

    Weight uncertainty in neural network

    Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International conference on machine learning, pages 1613--1622. PMLR, 2015

    2015

  7. [7]

    Credal Bayesian deep learning

    Michele Caprio, Souradeep Dutta, Kuk Jin Jang, Vivian Lin, Radoslav Ivanov, Oleg Sokolsky, and Insup Lee. Credal Bayesian deep learning. Transactions on Machine Learning Research, 2024. ISSN 2835-8856

    2024

  8. [8]

    u gner, and Stephan G \

    Bertrand Charpentier, Daniel Z \"u gner, and Stephan G \"u nnemann. Posterior network: Uncertainty estimation without OOD samples via density-based pseudo-counts. Advances in neural information processing systems, 33: 0 1356--1367, 2020

    2020

  9. [9]

    Integral imprecise probability metrics

    Siu Lun Chau, Michele Caprio, and Krikamol Muandet. Integral imprecise probability metrics. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025 a

    2025

  10. [10]

    Credal two-sample tests of epistemic uncertainty

    Siu Lun Chau, Antonin Schrab, Arthur Gretton, Dino Sejdinovic, and Krikamol Muandet. Credal two-sample tests of epistemic uncertainty. In International Conference on Artificial Intelligence and Statistics, pages 127--135. PMLR, 2025 b

    2025

  11. [11]

    Quantifying epistemic predictive uncertainty in conformal prediction

    Siu Lun Chau, Soroush H Zargarbashi, Yusuf Sale, and Michele Caprio. Quantifying epistemic predictive uncertainty in conformal prediction. arXiv preprint arXiv:2602.01667, 2026

    arXiv 2026

  12. [12]

    Bayesian networks with imprecise probabilities: Theory and application to classification

    Giorgio Corani, Alessandro Antonucci, and Marco Zaffalon. Bayesian networks with imprecise probabilities: Theory and application to classification. Data Mining: Foundations and Intelligent Paradigms: Volume 1: Clustering, Association and Classification, pages 49--93, 2012

    2012

  13. [13]

    De Campos, Juan F

    Luis M. De Campos, Juan F. Huete, and Serafin Moral. Probability intervals: A tool for uncertain reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 02 0 (02): 0 167--196, June 1994. ISSN 0218-4885, 1793-6411. doi:10.1142/S0218488594000146

    work page doi:10.1142/s0218488594000146 1994

  14. [14]

    Representing partial ignorance

    Didier Dubois, Henri Prade, and Philippe Smets. Representing partial ignorance. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 26 0 (3): 0 361--377, 2002

    2002

  15. [15]

    Masksembles for uncertainty estimation

    Nikita Durasov, Timur Bagautdinov, Pierre Baque, and Pascal Fua. Masksembles for uncertainty estimation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 13539--13548, 2021

    2021

  16. [16]

    Dropout as a Bayesian approximation: Representing model uncertainty in deep learning

    Yarin Gal and Zoubin Ghahramani. Dropout as a Bayesian approximation: Representing model uncertainty in deep learning. In International Conference on Machine Learning, pages 1050--1059. PMLR, 2016

    2016

  17. [17]

    Practical variational inference for neural networks

    Alex Graves. Practical variational inference for neural networks. Advances in neural information processing systems, 24, 2011

    2011

  18. [18]

    Transmission of information 1

    Ralph VL Hartley. Transmission of information 1. Bell System technical journal, 7 0 (3): 0 535--563, 1928

    1928

  19. [19]

    Deep residual learning for image recognition

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770--778, 2016

    2016

  20. [20]

    u hn and Eyke H \

    Jens Christian H \"u hn and Eyke H \"u llermeier. FR3 : A fuzzy rule learner for inducing reliable classifiers. IEEE Transactions on Fuzzy Systems, 17 0 (1): 0 138--149, 2008

    2008

  21. [21]

    Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods

    Eyke H \"u llermeier and Willem Waegeman. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine learning, 110 0 (3): 0 457--506, 2021

    2021

  22. [22]

    Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison

    Eyke H \"u llermeier, S \'e bastien Destercke, and Mohammad Hossein Shaker. Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison. In Uncertainty in Artificial Intelligence, pages 548--557. PMLR, 2022

    2022

  23. [23]

    Paul F Jaeger, Carsten Tim L \"u th, Lukas Klein, and Till J. Bungert. A call to reflect on evaluation practices for failure detection in image classification. In The Eleventh International Conference on Learning Representations, 2023

    2023

  24. [24]

    Hands-on Bayesian neural networks—a tutorial for deep learning users

    Laurent Valentin Jospin, Hamid Laga, Farid Boussaid, Wray Buntine, and Mohammed Bennamoun. Hands-on Bayesian neural networks—a tutorial for deep learning users. IEEE Computational Intelligence Magazine, 17 0 (2): 0 29--48, 2022

    2022

  25. [25]

    A generalized deep learning framework for whole-slide image segmentation and analysis

    Mahendra Khened, Avinash Kori, Haran Rajkumar, Ganapathy Krishnamurthi, and Balaji Srinivasan. A generalized deep learning framework for whole-slide image segmentation and analysis. Scientific reports, 11 0 (1): 0 11579, 2021

    2021

  26. [26]

    CIFAR-10 (Canadian Institute For Advanced Research)

    Alex Krizhevsky, Vinod Nair, and Geoffrey Hinton. CIFAR-10 (Canadian Institute For Advanced Research) . Technical report, CIFAR, 2009

    2009

  27. [27]

    Bayesian hypernetworks

    David Krueger, Chin-Wei Huang, Riashat Islam, Ryan Turner, Alexandre Lacoste, and Aaron Courville. Bayesian hypernetworks. ArXiv Preprint ArXiv:1710.04759, 2017

    Pith/arXiv arXiv 2017

  28. [28]

    Simple and scalable predictive uncertainty estimation using deep ensembles

    Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in Neural Information Processing Systems, 30, 2017

    2017

  29. [29]

    Packed ensembles for efficient uncertainty estimation

    Olivier Laurent, Adrien Lafage, Enzo Tartaglione, Geoffrey Daniel, Jean-marc Martinez, Andrei Bursuc, and Gianni Franchi. Packed ensembles for efficient uncertainty estimation. In The Eleventh International Conference on Learning Representations, 2022

    2022

  30. [30]

    The enterprise of knowledge: An essay on knowledge, credal probability, and chance

    Isaac Levi. The enterprise of knowledge: An essay on knowledge, credal probability, and chance. MIT press, 1980

    1980

  31. [31]

    Position: Supervised classifiers answer the wrong questions for ood detection

    Yucen Lily Li, Daohan Lu, Polina Kirichenko, Shikai Qiu, Tim GJ Rudner, C Bayan Bruss, and Andrew Gordon Wilson. Position: Supervised classifiers answer the wrong questions for ood detection. In Forty-second International Conference on Machine Learning Position Paper Track, 2025

    2025

  32. [32]

    o hr, Paul Hofman, Felix Mohr, and Eyke H \

    Timo L \"o hr, Paul Hofman, Felix Mohr, and Eyke H \"u llermeier. Credal prediction based on relative likelihood. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025

    2025

  33. [33]

    Predictive uncertainty estimation via prior networks

    Andrey Malinin and Mark Gales. Predictive uncertainty estimation via prior networks. Advances in neural information processing systems, 31, 2018

    2018

  34. [34]

    A unified evaluation framework for epistemic predictions

    Shireen Kudukkil Manchingal, Muhammad Mubashar, Kaizheng Wang, and Fabio Cuzzolin. A unified evaluation framework for epistemic predictions. In The 28th International Conference on Artificial Intelligence and Statistics, 2025 a

    2025

  35. [35]

    Random-set neural networks

    Shireen Kudukkil Manchingal, Muhammad Mubashar, Kaizheng Wang, Keivan Shariatmadar, and Fabio Cuzzolin. Random-set neural networks. In The Thirteenth International Conference on Learning Representations, 2025 b

    2025

  36. [36]

    Benchmarking common uncertainty estimation methods with histopathological images under domain shift and label noise

    Hendrik A Mehrtens, Alexander Kurz, Tabea-Clara Bucher, and Titus J Brinker. Benchmarking common uncertainty estimation methods with histopathological images under domain shift and label noise. Medical image analysis, 89: 0 102914, 2023

    2023

  37. [37]

    Dropconnect is effective in modeling uncertainty of Bayesian deep networks

    Aryan Mobiny, Pengyu Yuan, Supratik K Moulik, Naveen Garg, Carol C Wu, and Hien Van Nguyen. Dropconnect is effective in modeling uncertainty of Bayesian deep networks. Scientific Reports, 11 0 (1): 0 1--14, 2021

    2021

  38. [38]

    Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks

    B \'a lint Mucs \'a nyi, Michael Kirchhof, and Seong Joon Oh. Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. In The Thirty-eight Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2024

    2024

  39. [39]

    Deep deterministic uncertainty: A new simple baseline

    Jishnu Mukhoti, Andreas Kirsch, Joost van Amersfoort, Philip HS Torr, and Yarin Gal. Deep deterministic uncertainty: A new simple baseline. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 24384--24394, 2023

    2023

  40. [40]

    Second-order uncertainty quantification: A distance-based approach

    Yusuf Sale, Viktor Bengs, Michele Caprio, and Eyke H \"u llermeier. Second-order uncertainty quantification: A distance-based approach. In Forty-first International Conference on Machine Learning, 2024 a

    2024

  41. [41]

    o hr, Lisa Wimmer, Thomas Nagler, and Eyke H \

    Yusuf Sale, Paul Hofman, Timo L \"o hr, Lisa Wimmer, Thomas Nagler, and Eyke H \"u llermeier. Label-wise aleatoric and epistemic uncertainty quantification. In The Fortieth Conference on Uncertainty in Artificial Intelligence, 2024 b

    2024

  42. [42]

    Ensemble-based uncertainty quantification: Bayesian versus credal inference

    Mohammad Hossein Shaker and Eyke H \"u llermeier. Ensemble-based uncertainty quantification: Bayesian versus credal inference. In Proceedings of the Tirty-first Workshop Computational Intelligence, volume 25, page 63, 2021

    2021

  43. [43]

    Uncertainty quantification for Bayesian optimization

    Rui Tuo and Wenjia Wang. Uncertainty quantification for Bayesian optimization. In Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera, editors, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, volume 151 of Proceedings of Machine Learning Research, pages 2862--2884. PMLR, 28--30 Mar 2022

    2022

  44. [44]

    Credal deep ensembles for uncertainty quantification

    Kaizheng Wang, Fabio Cuzzolin, Shireen Kudukkil Manchingal, Keivan Shariatmadar, David Moens, and Hans Hallez. Credal deep ensembles for uncertainty quantification. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024 a

    2024

  45. [45]

    Credal ensemble distillation for uncertainty quantification, 2025 a

    Kaizheng Wang, Fabio Cuzzolin, David Moens, and Hans Hallez. Credal ensemble distillation for uncertainty quantification, 2025 a

    2025

  46. [46]

    Credal wrapper of model averaging for uncertainty estimation in classification

    Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, and Hans Hallez. Credal wrapper of model averaging for uncertainty estimation in classification. In The Thirteenth International Conference on Learning Representations, 2025 b

    2025

  47. [47]

    CreINNs : Credal-set interval neural networks for uncertainty estimation in classification tasks

    Kaizheng Wang, Keivan Shariatmadar, Shireen Kudukkil Manchingal, Fabio Cuzzolin, David Moens, and Hans Hallez. CreINNs : Credal-set interval neural networks for uncertainty estimation in classification tasks. Neural Networks, 185: 0 107198, 2025 c . ISSN 0893-6080. doi:https://doi.org/10.1016/j.neunet.2025.107198

    arXiv 2025

  48. [48]

    A review of uncertainty representation and quantification in neural networks

    Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, and Hans Hallez. A review of uncertainty representation and quantification in neural networks. IEEE Transactions on Pattern Analysis and Machine Intelligence, 48 0 (3): 0 2476--2495, 2026 a . doi:10.1109/TPAMI.2025.3626645

    arXiv 2026

  49. [49]

    Learning credal ensembles via distributionally robust optimization

    Kaizheng Wang, Ghifari Adam Faza, Fabio Cuzzolin, Siu Lun Chau, David Moens, and Hans Hallez. Learning credal ensembles via distributionally robust optimization. arXiv preprint arXiv:2602.08470, 2026 b

    Pith/arXiv arXiv 2026

  50. [50]

    Navigating the waters of object detection: Evaluating the robustness of real-time object detection models for autonomous surface vehicles

    Yunjia Wang, Kaizheng Wang, Zihao Zhang, Jeroen Boydens, Davy Pissoort, and Mathias Verbeke. Navigating the waters of object detection: Evaluating the robustness of real-time object detection models for autonomous surface vehicles. In 2024 IEEE conference on artificial intelligence (CAI), pages 985--992. IEEE, 2024 b

    2024

  51. [51]

    Enhancing the dependability of autonomous surface vehicles through robustness benchmarking of real-time object detection models

    Yunjia Wang, Zihao Zhang, Kaizheng Wang, Holger Caesar, Jeroen Boydens, Davy Pissoort, and Mathias Verbeke. Enhancing the dependability of autonomous surface vehicles through robustness benchmarking of real-time object detection models. Expert Systems with Applications, page 129151, 2025 d

    2025

  52. [52]

    Batchensemble: an alternative approach to efficient ensemble and lifelong learning

    Yeming Wen, Dustin Tran, and Jimmy Ba. Batchensemble: an alternative approach to efficient ensemble and lifelong learning. arXiv preprint arXiv:2002.06715, 2020

    Pith/arXiv arXiv 2002

  53. [53]

    Bayesian deep learning and a probabilistic perspective of generalization

    Andrew G Wilson and Pavel Izmailov. Bayesian deep learning and a probabilistic perspective of generalization. Advances in neural information processing systems, 33: 0 4697--4708, 2020

    2020

  54. [54]

    Learning from the wisdom of crowds by minimax entropy

    Dengyong Zhou, Sumit Basu, Yi Mao, and John Platt. Learning from the wisdom of crowds by minimax entropy. Advances in Neural Information Processing Systems, 25, 2012

    2012