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

cs.LGFeb 26, 2026

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

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 & 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 $\pm$ 0.0568	-11	-18	0.9637 $\pm$ 0.0176	-12	-19	0.9887 $\pm$ 0.0156	-12	-18	0.9807 $\pm$ 0.0050	-12	-18
LWV	0.9625 $\pm$ 0.0532	0	-5	0.9669 $\pm$ 0.0131	0	-6	0.9899 $\pm$ 0.0151	0	-4	0.9815 $\pm$ 0.0049	0	-6
WD	0.9633 $\pm$ 0.0530	11	21	0.9683 $\pm$ 0.0125	12	30	0.9904 $\pm$ 0.0144	12	25	0.9823 $\pm$ 0.0046	12	28
GH	0.9633 $\pm$ 0.0523	6	15	0.9680 $\pm$ 0.0126	10	12	0.9903 $\pm$ 0.0144	12	19	0.9822 $\pm$ 0.0046	12	20
$H_{\text{diff}}$	0.9594 $\pm$ 0.0517	-12	-28	0.9613 $\pm$ 0.0164	-12	-29	0.9848 $\pm$ 0.0237	-12	-30	0.9764 $\pm$ 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 $\pm$ 0.0840	-3	-6	0.8250 $\pm$ 0.0859	-5	-11	0.7657 $\pm$ 0.0301	-8	-25	0.8639 $\pm$ 0.0276	0	-11
LWV	0.8655 $\pm$ 0.0797	-9	-15	0.8300 $\pm$ 0.0776	-7	-13	0.7694 $\pm$ 0.0295	-4	-21	0.8466 $\pm$ 0.0256	-12	-30
WD	0.8822 $\pm$ 0.0739	12	19	0.8512 $\pm$ 0.0720	12	18	0.8174 $\pm$ 0.0227	12	18	0.8731 $\pm$ 0.0232	12	10
GH	0.8856 $\pm$ 0.0720	12	29	0.8561 $\pm$ 0.0703	12	30	0.8303 $\pm$ 0.0197	12	30	0.8801 $\pm$ 0.0226	9	27
$H_{\mathrm{diff}}$	0.8516 $\pm$ 0.0893	-12	-30	0.7929 $\pm$ 0.0912	-12	-30	0.7797 $\pm$ 0.0343	-12	-8	0.8744 $\pm$ 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.

In-distribution Camelyon17 test data
	DE	PackEns	MaskEns	BatchEns	SVI	MCDO	Overall
MI	0.9737 $\pm$ 0.0024	0.9725 $\pm$ 0.0021	0.9722 $\pm$ 0.0033	0.9742 $\pm$ 0.0033	0.9643 $\pm$ 0.0161	0.9041 $\pm$ 0.1289	0.9602 $\pm$ 0.0568
LWV	0.9749 $\pm$ 0.0020	0.9750 $\pm$ 0.0020	0.9748 $\pm$ 0.0028	0.9765 $\pm$ 0.0025	0.9655 $\pm$ 0.0139	0.9082 $\pm$ 0.1195	0.9625 $\pm$ 0.0532
WD	0.9756 $\pm$ 0.0018	0.9754 $\pm$ 0.0020	0.9753 $\pm$ 0.0028	0.9770 $\pm$ 0.0025	0.9659 $\pm$ 0.0139	0.9106 $\pm$ 0.1200	0.9633 $\pm$ 0.0530
GH	0.9754 $\pm$ 0.0019	0.9754 $\pm$ 0.0020	0.9753 $\pm$ 0.0028	0.9769 $\pm$ 0.0026	0.9656 $\pm$ 0.0144	0.9111 $\pm$ 0.1183	0.9633 $\pm$ 0.0523
$H_{\mathrm{diff}}$	0.9726 $\pm$ 0.0020	0.9705 $\pm$ 0.0026	0.9708 $\pm$ 0.0036	0.9731 $\pm$ 0.0035	0.9616 $\pm$ 0.0179	0.9080 $\pm$ 0.1164	0.9594 $\pm$ 0.0517

Appendix

Table A.2 : AUARC scores for selective prediction tasks on various datasets.

SeaShip v.s. SeaShip-O
	DE	PackEns	MaskEns	BatchEns	SVI	MCDO	Overall
MI	0.9153 $\pm$ 0.0026	0.9347 $\pm$ 0.0127	0.9145 $\pm$ 0.0106	0.9072 $\pm$ 0.0139	0.7129 $\pm$ 0.0678	0.8315 $\pm$ 0.0379	0.8694 $\pm$ 0.0840
LWV	0.9085 $\pm$ 0.0021	0.9265 $\pm$ 0.0136	0.9066 $\pm$ 0.0107	0.8986 $\pm$ 0.0139	0.7147 $\pm$ 0.0672	0.8378 $\pm$ 0.0355	0.8655 $\pm$ 0.0797
WD	0.9235 $\pm$ 0.0028	0.9373 $\pm$ 0.0136	0.9202 $\pm$ 0.0108	0.9129 $\pm$ 0.0128	0.7449 $\pm$ 0.0681	0.8545 $\pm$ 0.0322	0.8822 $\pm$ 0.0739
GH	0.9240 $\pm$ 0.0025	0.9403 $\pm$ 0.0125	0.9235 $\pm$ 0.0104	0.9161 $\pm$ 0.0126	0.7519 $\pm$ 0.0654	0.8577 $\pm$ 0.0314	0.8856 $\pm$ 0.0720
$H_{\mathrm{diff}}$	0.9023 $\pm$ 0.0077	0.9220 $\pm$ 0.0159	0.8939 $\pm$ 0.0136	0.8913 $\pm$ 0.0174	0.6841 $\pm$ 0.0725	0.8162 $\pm$ 0.0397	0.8516 $\pm$ 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
$H_{\mathrm{diff}}$	-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
$H_{\mathrm{diff}}$	-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 $\pm 1$ 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 $\pm 1$ 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 $\pm 1$ 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 $\pm 1$ 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 $\pm$ 0.0017	-12	-18	0.9148 $\pm$ 0.0713	-1	-8	0.9074 $\pm$ 0.0701	6	3
LWV	0.9910 $\pm$ 0.0016	0	-6	0.9026 $\pm$ 0.0604	-9	-27	0.8943 $\pm$ 0.0609	-10	-28
WD	0.9912 $\pm$ 0.0015	12	25	0.9323 $\pm$ 0.0511	10	14	0.9126 $\pm$ 0.0536	4	-2
GH	0.9912 $\pm$ 0.0015	12	23	0.9414 $\pm$ 0.0473	12	30	0.9173 $\pm$ 0.0523	1	12
$H_{\text{diff}}$	0.9874 $\pm$ 0.0026	-12	-30	0.9215 $\pm$ 0.0612	-7	-4	0.9306 $\pm$ 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 $\pm$ 0.0007	-10	-16	0.9704 $\pm$ 0.0108	6	13	0.9765 $\pm$ 0.0074	6	12
LWV	0.9952 $\pm$ 0.0006	2	5	0.9570 $\pm$ 0.0122	-12	-30	0.9629 $\pm$ 0.0097	-12	-28
WD	0.9953 $\pm$ 0.0007	8	22	0.9704 $\pm$ 0.0092	6	12	0.9769 $\pm$ 0.0062	6	12
GH	0.9953 $\pm$ 0.0007	10	15	0.9741 $\pm$ 0.0082	12	29	0.9806 $\pm$ 0.0051	12	30
$H_{\text{diff}}$	0.9944 $\pm$ 0.0010	-12	-30	0.9649 $\pm$ 0.0118	-9	-15	0.9653 $\pm$ 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

\pm 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

\pm 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

\pm 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

\pm 1

standard deviation, and faint lines show individual runs. Mean AUROC values are reported in the legend.

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Evaluating Epistemic Uncertainty: Beyond OOD Detection and Active Learning

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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.

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