Set-based v.s. Distribution-based Representations of Epistemic Uncertainty: A Comparative Study
Authors: Kaizheng Wang, Yunjia Wang, Fabio Cuzzolin, David Moens, Hans Hallez, Siu Lun Chau
Organizations: College of Computing and Data Science, Nanyang Technological University, Singapore · Department of Computer Science, KU Leuven, Belgium · School of Engineering, Computing, and Mathematics, Oxford Brookes University, U.K. · Department of Mechanical Engineering, KU Leuven, Belgium
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 & tables
Figure 1: Illustration of our comparative study framework.
In-distribution Camelyon17
Distribution-shift Camelyon17
In-distribution SeaShip
In-distribution CIFAR10
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
MI
0.9602 ± 0.0568
-11
-18
0.9637 ± 0.0176
-12
-19
0.9887 ± 0.0156
-12
-18
0.9807 ± 0.0050
-12
-18
LWV
0.9625 ± 0.0532
0
-5
0.9669 ± 0.0131
0
-6
0.9899 ± 0.0151
0
-4
0.9815 ± 0.0049
0
-6
WD
0.9633 ± 0.0530
11
21
0.9683 ± 0.0125
12
30
0.9904 ± 0.0144
12
25
0.9823 ± 0.0046
12
28
GH
0.9633 ± 0.0523
6
15
0.9680 ± 0.0126
10
12
0.9903 ± 0.0144
12
19
0.9822 ± 0.0046
12
20
Hdiff
0.9594 ± 0.0517
-12
-28
0.9613 ± 0.0164
-12
-29
0.9848 ± 0.0237
-12
-30
0.9764 ± 0.0068
-12
-30
Table 1: Average AUARC on the selective prediction task, computed per dataset and averaged across six predictive model architectures. We additionally report net wins, obtained from pairwise comparisons both within the same uncertainty-representation category (intra-representation) and across different categories (inter-representation). For both AUARC and net wins, higher values indicate better performance. In each column, the best-performing method is highlighted in bold red and the second-best in bold blue . Detailed AUARC score as well as the intra- and inter-representation comparisons for each underlying predictive model are provided in Tables A.3 , A.4 , and A.5 , respectively.
SeaShip v.s. SeaShip-O
SeaShip v.s. SeaShip-C
CIFAR10 v.s. SVHN
CIFAR10 v.s. FMNIST
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
MI
0.8694 ± 0.0840
-3
-6
0.8250 ± 0.0859
-5
-11
0.7657 ± 0.0301
-8
-25
0.8639 ± 0.0276
0
-11
LWV
0.8655 ± 0.0797
-9
-15
0.8300 ± 0.0776
-7
-13
0.7694 ± 0.0295
-4
-21
0.8466 ± 0.0256
-12
-30
WD
0.8822 ± 0.0739
12
19
0.8512 ± 0.0720
12
18
0.8174 ± 0.0227
12
18
0.8731 ± 0.0232
12
10
GH
0.8856 ± 0.0720
12
29
0.8561 ± 0.0703
12
30
0.8303 ± 0.0197
12
30
0.8801 ± 0.0226
9
27
Hdiff
0.8516 ± 0.0893
-12
-30
0.7929 ± 0.0912
-12
-30
0.7797 ± 0.0343
-12
-8
0.8744 ± 0.0266
3
17
Table 2: Average AUROC on the OOD detection task, computed per dataset pair and averaged across six predictive model architectures. We additionally report net wins, obtained from pairwise comparisons both within the same uncertainty-representation category (intra-representation) and across different categories (inter-representation). For both AUROC and net wins, higher values indicate better performance. In each column, the best-performing method is highlighted in bold red and the second-best in bold blue . Detailed AUROC score as well as the intra- and inter-representation net-win comparisons for each underlying predictive model are provided in Tables A.3 , A.6 , and A.7 , respectively.
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 blue (distribution-based measures) and orange (credal-based measures), while inter-representation comparisons are shown in green.
Appendix figures & tables29 assets
Supplementary material from the paper’s appendix.
Appendix
Figure A.1 : An example of the whole slide image (referring to node 2 of patient 017) with ground-truth annotations.
ID dataset
Domain-shifted test dataset
Training
Validation
Testing
Center 2
Center 4
383406
110561
109060
89351
166607
Appendix
Table A.1 : Number of tile instances in each dataset split.
Figure A.2 : Examples of the SeaShip dataset v.s. instances from the Seaship-C at severity level 3.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Overall
In-distribution Camelyon17 test data
MI
0.9737 ± 0.0024
0.9725 ± 0.0021
0.9722 ± 0.0033
0.9742 ± 0.0033
0.9643 ± 0.0161
0.9041 ± 0.1289
0.9602 ± 0.0568
LWV
0.9749 ± 0.0020
0.9750 ± 0.0020
0.9748 ± 0.0028
0.9765 ± 0.0025
0.9655 ± 0.0139
0.9082 ± 0.1195
0.9625 ± 0.0532
WD
0.9756 ± 0.0018
0.9754 ± 0.0020
0.9753 ± 0.0028
0.9770 ± 0.0025
0.9659 ± 0.0139
0.9106 ± 0.1200
0.9633 ± 0.0530
GH
0.9754 ± 0.0019
0.9754 ± 0.0020
0.9753 ± 0.0028
0.9769 ± 0.0026
0.9656 ± 0.0144
0.9111 ± 0.1183
0.9633 ± 0.0523
Hdiff
0.9726 ± 0.0020
0.9705 ± 0.0026
0.9708 ± 0.0036
0.9731 ± 0.0035
0.9616 ± 0.0179
0.9080 ± 0.1164
0.9594 ± 0.0517
Appendix
Table A.2 : AUARC scores for selective prediction tasks on various datasets.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Overall
SeaShip v.s. SeaShip-O
MI
0.9153 ± 0.0026
0.9347 ± 0.0127
0.9145 ± 0.0106
0.9072 ± 0.0139
0.7129 ± 0.0678
0.8315 ± 0.0379
0.8694 ± 0.0840
LWV
0.9085 ± 0.0021
0.9265 ± 0.0136
0.9066 ± 0.0107
0.8986 ± 0.0139
0.7147 ± 0.0672
0.8378 ± 0.0355
0.8655 ± 0.0797
WD
0.9235 ± 0.0028
0.9373 ± 0.0136
0.9202 ± 0.0108
0.9129 ± 0.0128
0.7449 ± 0.0681
0.8545 ± 0.0322
0.8822 ± 0.0739
GH
0.9240 ± 0.0025
0.9403 ± 0.0125
0.9235 ± 0.0104
0.9161 ± 0.0126
0.7519 ± 0.0654
0.8577 ± 0.0314
0.8856 ± 0.0720
Hdiff
0.9023 ± 0.0077
0.9220 ± 0.0159
0.8939 ± 0.0136
0.8913 ± 0.0174
0.6841 ± 0.0725
0.8162 ± 0.0397
0.8516 ± 0.0893
Appendix
Table A.3 : AUROC scores for OOD detection tasks on various dataset pairs.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
Distribution-based measures
In-distribution Camelyon17 test data
Distribution-shift Camelyon17 test data
MI
-2
-2
-2
-2
-1
-2
-11
-2
-2
-2
-2
-2
-2
-12
LWV
0
0
0
0
0
0
0
0
0
0
0
0
0
0
WD
2
2
2
2
1
2
11
2
2
2
2
2
2
12
In-distribution SeaShip test data
In-distribution CIFAR10 test data
Appendix
Table A.4: Net wins for intra-representation comparisons on selective prediction tasks across datasets. The first rank according to net wins is shown in bold red, and the second rank in bold blue.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
In-distribution Camelyon17 test data
Distribution-shift Camelyon17 test data
MI
-3
-3
-3
-3
-2
-4
-18
-3
-3
-3
-3
-3
-4
-19
LWV
-1
-1
-1
-1
1
-2
-5
-1
-1
-1
-1
-1
-1
-6
WD
5
3
3
5
2
3
21
5
5
5
5
5
5
30
GH
2
3
3
2
2
3
15
2
2
2
2
2
2
12
Hdiff
-5
-5
-5
-5
-5
-3
-28
-5
-5
-5
-5
-5
-4
-29
Appendix
Table A.5: Net wins for inter-representation comparisons on selective prediction tasks across datasets. The first rank according to net wins is shown in bold red, and the second rank in bold blue.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
Distribution-based measures
SeaShip v.s. SeaShip-O
SeaShip v.s. SeaShip-C
MI
0
0
0
0
-1
-2
-3
0
0
-1
0
-2
-2
-5
LWV
-2
-2
-2
-2
-1
0
-9
-2
-2
-1
-2
0
0
-7
WD
2
2
2
2
2
2
12
2
2
2
2
2
2
12
CIFAR10 v.s. SVHN
CIFAR10 v.s. FMNIST
Appendix
Table A.6: Net wins for intra-representation comparisons on OOD detection tasks across datasets. The first rank according to net wins is shown in bold red, and the second rank in bold blue.
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
DE
PackEns
MaskEns
BatchEns
SVI
MCDO
Total
SeaShip v.s. SeaShip-O
SeaShip v.s. SeaShip-C
MI
-1
0
0
0
-2
-3
-6
-1
-1
-2
-1
-3
-3
-11
LWV
-3
-3
-3
-3
-2
-1
-15
-3
-3
-2
-3
-1
-1
-13
WD
4
3
3
3
3
3
19
3
3
3
3
3
3
18
GH
4
5
5
5
5
5
29
5
5
5
5
5
5
30
Hdiff
-5
-5
-5
-5
-5
-5
-30
-5
-5
-5
-5
-5
-5
-30
Appendix
Table A.7: Net wins for inter-representation comparisons on OOD detection tasks across datasets. The first rank according to net wins is shown in bold red, and the second rank in bold blue.
Figure A.3 : Statistical significance plots on different selective prediction (a-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 blue (distribution-based measures) and orange (credal-based measures), while inter-representation comparisons are shown in green.
Figure A.4 : Statistical significance plots on different OOD detection (e-h) 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 blue (distribution-based measures) and orange (credal-based measures), while inter-representation comparisons are shown in green.
Figure A.5 : Accuracy rejection curves using different uncertainty representations and measures on in-distribution Camelyon17 test data across different underlying predictive models.
Figure A.6 : Accuracy rejection curves using different uncertainty representations and measures on distribution-shift Camelyon17 test data across different underlying predictive models.
Figure A.7 : Accuracy rejection curves using different uncertainty representations and measures on in-distribution SeaShip test data across different underlying predictive models.
Figure A.8 : Accuracy rejection curves using different uncertainty representations and measures on in-distribution CIFAR10 test data across different underlying predictive models.
Figure A.9 : ROC curves for OOD detection (SeaShip v.s. Seachip-O) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.10 : ROC curves for OOD detection (SeaShip v.s. Seachip-C) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.11 : ROC curves for OOD detection (CIFAR10 v.s. SVHN) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.12 : ROC curves for OOD detection (CIFAR10 v.s. FMNIST) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Selective Prediction
OOD Detection (SVHN)
OOD Detection (CIFAR10)
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
MI
0.9901 ± 0.0017
-12
-18
0.9148 ± 0.0713
-1
-8
0.9074 ± 0.0701
6
3
LWV
0.9910 ± 0.0016
0
-6
0.9026 ± 0.0604
-9
-27
0.8943 ± 0.0609
-10
-28
WD
0.9912 ± 0.0015
12
25
0.9323 ± 0.0511
10
14
0.9126 ± 0.0536
4
-2
GH
0.9912 ± 0.0015
12
23
0.9414 ± 0.0473
12
30
0.9173 ± 0.0523
1
12
Hdiff
0.9874 ± 0.0026
-12
-30
0.9215 ± 0.0612
-7
-4
0.9306 ± 0.0554
11
29
Appendix
Table A.8 : Average AUARC on the selective prediction task and AUROC on OOD detection tasks for models trained on FMNIST , averaged across six distinct predictive models. Intra- and inter-representation net wins are reported alongside. The first rank is shown in bold red , the second in bold blue .
Figure A.13 : Statistical significance plots on different selective prediction (a) and OOD detection (b-c) benchmarks across different underlying predictive models trained on the FMNIST dataset. 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 blue (distribution-based measures) and orange (credal-based measures), while inter-representation comparisons are shown in green.
Selective Prediction
OOD Detection (FMNIST)
OOD Detection (CIFAR10)
Average scores
Intra
Inter
Average scores
Intra
Inter
Average scores
Intra
Inter
MI
0.9950 ± 0.0007
-10
-16
0.9704 ± 0.0108
6
13
0.9765 ± 0.0074
6
12
LWV
0.9952 ± 0.0006
2
5
0.9570 ± 0.0122
-12
-30
0.9629 ± 0.0097
-12
-28
WD
0.9953 ± 0.0007
8
22
0.9704 ± 0.0092
6
12
0.9769 ± 0.0062
6
12
GH
0.9953 ± 0.0007
10
15
0.9741 ± 0.0082
12
29
0.9806 ± 0.0051
12
30
Hdiff
0.9944 ± 0.0010
-12
-30
0.9649 ± 0.0118
-9
-15
0.9653 ± 0.0100
-12
-20
Appendix
Table A.9 : Average AUARC on the selective prediction task and AUROC on OOD detection tasks for models trained on SVHN , averaged across six distinct predictive models. Intra- and inter-representation net wins are reported alongside. The first rank is shown in bold red , the second in bold blue .
Figure A.14 : Statistical significance plots on different selective prediction (a) and OOD detection (b-c) benchmarks across different underlying predictive models trained on the SVHN dataset. 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 blue (distribution-based measures) and orange (credal-based measures), while inter-representation comparisons are shown in green.
Figure A.15 : Accuracy rejection curves using different uncertainty representations and measures on in-distribution FMNIST test data across different underlying predictive models.
Figure A.16 : ROC curves for OOD detection ( ID: FMNIST v.s. OOD: SVHN ) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.17 : ROC curves for OOD detection ( ID: FMNIST v.s. OOD: CIFAR10 ) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.18 : Accuracy rejection curves using different uncertainty representations and measures on in-distribution SVHN test data across different underlying predictive models.
Figure A.19 : ROC curves for OOD detection ( ID: SVHN v.s. OOD: FMNIST ) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Figure A.20 : ROC curves for OOD detection ( ID: SVHN v.s. OOD: CIFAR10 ) across uncertainty measures and backbone methods. Each panel corresponds to one backbone, with all uncertainty measures plotted together. Solid lines represent the mean TPR over 10 independent runs, shaded regions indicate ±1 standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.
Current evaluation of epistemic uncertainty relies on tasks such as out-ofdistribution detection and active learning. However, the Bayes-optimal decision strategies for these tasks do not coincide with the scores commonly used to quantify epistemic uncertainty. Building on the epistemic reject-option framework, we evaluate epistemic uncertainty using its ability to identify regret, the reducible error. Formulating selective prediction as a constrained optimization over coverage, expected risk, and regret, we prove the optimal selector is a thresholded convex combination of the ground-truth aleatoric and epistemic uncertainties. This theoretical unification exposes a weakness in recent uncertainty disentanglement literature: we demonstrate that standard correlation metrics between learned components do not necessarily predict their actual operational utility. We instead propose to evaluate the achievable risk, regret, coverage surface of the decomposition as a diagnostic for joint disentanglement and utility. Benchmarking standard methods on datasets with dense human annotations reveals that decision-theoretic rankings can disagree substantially with proxy-task rankings, including pairwise rank inversions between methods that are top-ranked on one criterion and bottom-ranked on other.
Jakub Paplhám, Willem Waegeman, Eyke Hüllermeier +1
Czech Technical University in Prague · Ghent University · LMU Munich, MCML, DFKI
Uncertainty estimation is critical for deploying machine learning models in high-stakes settings. However, classical calibration only assesses the reliability of predicted probabilities and does not evaluate whether epistemic uncertainty estimates are themselves trustworthy. This limitation is particularly relevant for second-order classification models. We introduce epistemic calibration, a principled criterion that measures whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. We show that epistemic calibration is a strictly stronger notion than classical calibration and captures failure modes invisible to standard metrics. We relate this work to the existing literature through an impossibility theorem that holds under the epistemic calibration hypothesis. To operationalize this concept, we propose the Expected Epistemic Calibration Error (EECE), which we prove to be a consistent estimator of a True Epistemic Calibration Error (TECE). Experiments across a broad range of uncertainty quantification methods show that epistemic calibration is a coherent and meaningful criterion and reveal substantial differences across methods, despite similar predictive performance.
Arthur Hoarau
Université de Lorraine, CentraleSupélec · Loria, CNRS, Metz, France
Credal sets, i.e., closed convex sets of probability measures, provide a natural framework to represent aleatoric and epistemic uncertainty in machine learning. Yet how to quantify these two types of uncertainty for a given credal set, particularly in multiclass classification, remains underexplored. In this paper, we propose a distance-based approach to quantify total, aleatoric, and epistemic uncertainty for credal sets. Concretely, we introduce a family of such measures within the framework of Integral Probability Metrics (IPMs). The resulting quantities admit clear semantic interpretations, satisfy natural theoretical desiderata, and remain computationally tractable for common choices of IPMs. We instantiate the framework with the total variation distance and obtain simple, efficient uncertainty measures for multiclass classification. In the binary case, this choice recovers established uncertainty measures, for which a principled multiclass generalization has so far been missing. Empirical results confirm practical usefulness, with favorable performance at low computational cost.
Xabier Gonzalez-Garcia, Siu Lun Chau, Julian Rodemann +6
Dept. of Statistics, CS & Mathematics, Public Univ. of Navarre · EPIC Lab, Nanyang Technological Univ. Singapore · Rational Intelligence Lab, CISPA Helmholtz Center for Information Security +7