CUPID: A Plug-in Framework for Joint Aleatoric and Epistemic Uncertainty Estimation with a Single Model

Xinran Xu; Xiuyi Fan

cs.LGMar 11, 2026

CUPID: A Plug-in Framework for Joint Aleatoric and Epistemic Uncertainty Estimation with a Single Model

Authors: Xinran Xu, Xiuyi Fan

Organizations: Lee Kong Chian School of Medicine, Nanyang Technological University, Singapore · College of Computing and Data Science, Nanyang Technological University, Singapore · Centre for Medical Technologies & Innovations, National Health Group, Singapore

Abstract

Accurate estimation of uncertainty in deep learning is critical for deploying models in high-stakes domains such as medical diagnosis and autonomous decision-making, where overconfident predictions can lead to harmful outcomes. In practice, understanding the reason behind a model's uncertainty and the type of uncertainty it represents can support risk-aware decisions, enhance user trust, and guide additional data collection. However, many existing methods only address a single type of uncertainty or require modifications and retraining of the base model, making them difficult to adopt in real-world systems. We introduce CUPID (Comprehensive Uncertainty Plug-in estImation moDel), a general-purpose module that jointly estimates aleatoric and epistemic uncertainty without modifying or retraining the base model. CUPID can be flexibly inserted into any layer of a pretrained network. It models aleatoric uncertainty through a learned Bayesian identity mapping and captures epistemic uncertainty by analyzing the model's internal responses to structured perturbations. We evaluate CUPID across a range of tasks, including classification, regression, and out-of-distribution detection. The results show that it consistently delivers competitive performance while offering layer-wise insights into the origins of uncertainty. By making uncertainty estimation modular, interpretable, and model-agnostic, CUPID supports more transparent and trustworthy AI. Related code and data are available at https://github.com/a-Fomalhaut-a/CUPID.

Figures & tables

Figure 1: CUPID uncertainty estimation on a 1D regression toy problem. CUPID is inserted into an MLP-based predictive model. CUPID captures both aleatoric (blue) and epistemic (red) uncertainty.

Figure 2: The CUPID pipeline. Aleatoric uncertainty is estimated using a dedicated Uncertainty Branch, while epistemic uncertainty is captured by measuring the variance between the original model output $\hat{\mathbf{y}}$ and the perturbed output $\hat{\mathbf{y}}^{\prime}$ .

Method	GLV2			HAM10000
Method	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
CUPID Alea.	0.870 ± 0.002	0.018 ± 0.001	0.941 ± 0.004	0.769 ± 0.023	0.067 ± 0.007	0.722 ± 0.014
CUPID Epis.	0.769 ± 0.015	0.034 ± 0.002	0.701 ± 0.051	0.855 ± 0.006	0.047 ± 0.001	0.907 ± 0.001
MC Dropout	0.768 ± 0.006	0.027 ± 0.001	0.888 ± 0.005	0.829 ± 0.001	0.076 ± 0.001	0.861 ± 0.002
Rate-in	0.815 ± 0.006	0.024 ± 0.001	0.816 ± 0.004	0.846 ± 0.001	0.048 ± 0.000	0.915 ± 0.000
IGRUE	0.642 ± 0.007	0.058 ± 0.002	0.199 ± 0.004	0.548 ± 0.004	0.157 ± 0.002	0.027 ± 0.018

Table 1: Performance of misclassification detection (misclassified samples as positive). The best model for each metric is in bold, and the second best is underlined. CUPID Aleatoric achieved the best performance on GLV2, while CUPID Epistemic performed best on HAM10000, suggesting different dominant sources of uncertainty across datasets.

Method	PAPILA		ACRIMA		CIFAR10
Method	AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )
CUPID Alea.	0.379 ± 0.027	0.333 ± 0.007	0.717 ± 0.029	0.661 ± 0.027	0.983 ± 0.005	0.998 ± 0.001
CUPID Epis.	0.877 ± 0.032	0.854 ± 0.027	0.978 ± 0.010	0.984 ± 0.007	0.898 ± 0.054	0.991 ± 0.005
MC Dropout	0.733 ± 0.002	0.586 ± 0.007	0.869 ± 0.003	0.816 ± 0.009	0.887 ± 0.004	0.986 ± 0.001
Rate-in	0.328 ± 0.005	0.329 ± 0.008	0.363 ± 0.003	0.390 ± 0.003	0.620 ± 0.001	0.927 ± 0.002
IGRUE	0.636 ± 0.114	0.486 ± 0.097	0.941 ± 0.008	0.944 ± 0.008	0.978 ± 0.005	0.998 ± 0.001

Table 2: Performance of OOD detection (OOD samples as positive). PAPILA and ACRIMA share the same research problem (glaucoma detection) with the ID dataset while CIFAR10 is a general classification dataset.

Method	Set5			Set14
Method	Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )	Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )
CUPID Alea.	0.528 ± 0.006	0.010 ± 0.000	0.045 ± 0.018	0.527 ± 0.002	0.012 ± 0.000	0.049 ± 0.005
CUPID Epis.	0.416 ± 0.004	0.018 ± 0.001	0.266 ± 0.007	0.449 ± 0.005	0.019 ± 0.000	0.226 ± 0.003
BayesCap	0.485 ± 0.038	0.010 ± 0.000	0.098 ± 0.001	0.422 ± 0.064	0.012 ± 0.000	0.100 ± 0.000
in-rotate	0.493 ± 0.000	0.010 ± 0.000	0.071 ± 0.000	0.490 ± 0.000	0.013 ± 0.000	0.072 ± 0.000
in-noise	0.370 ± 0.006	0.019 ± 0.000	0.051 ± 0.035	0.354 ± 0.001	0.022 ± 0.000	0.826 ± 0.006

Table 3: Performance on natural image datasets (Set5, Set14, BSDS100) and medical imaging dataset IXI (MRI scans). CUPID Aleatoric achieves the best results on the natural image benchmarks, while CUPID Epistemic performs best on the IXI dataset.

Figure 3: Comparison of visual results between error and uncertainty maps. CUPID Aleatoric shows the best texture alignment and highest correlation with error maps.

Figure 4: Performance of CUPID inserted at varying locations: misclassification detection (Left) and super-resolution (Right). Aleatoric uncertainty estimation improves when CUPID is placed closer to the output, while epistemic uncertainty benefits from earlier insertion points.

Method		PAPILA		ACRIMA		CIFAR10
Method		AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )
Max	Alea.	0.379 ± 0.027	0.333 ± 0.007	0.717 ± 0.029	0.661 ± 0.027	0.983 ± 0.005	0.998 ± 0.001
No max	Alea.	0.389 ± 0.026	0.338 ± 0.009	0.739 ± 0.042	0.696 ± 0.055	0.988 ± 0.003	0.999 ± 0.000
Max	Epis.	0.877 ± 0.032	0.854 ± 0.027	0.978 ± 0.010	0.984 ± 0.007	0.898 ± 0.054	0.991 ± 0.005
No max	Epis.	0.839 ± 0.017	0.790 ± 0.054	0.977 ± 0.006	0.982 ± 0.005	0.875 ± 0.024	0.989 ± 0.002

Table 4: Performance of differential feature loss on OOD task. ”No max” means remove

-\|\mathbf{m}_{l,n}-\mathbf{m}_{l,n}^{\prime}\|_{1}

in the loss function. Best-performing results for each metric are highlighted in bold.

Model	Aleatoric			Epistemic
Model	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
Joint	0.870 ± 0.002	0.018 ± 0.001	0.941 ± 0.004	0.769 ± 0.015	0.034 ± 0.002	0.701 ± 0.051
Seperate	0.863 ± 0.003	0.019 ± 0.001	0.899 ± 0.035	0.744 ± 0.017	0.043 ± 0.005	0.699 ± 0.014

Table 5: Misclassification detection performance on GLV2 (Joint vs. separate branches).

Model		PAPILA		ACRIMA		CIFAR10
Model		AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )
Alea.	Joint	0.379 ± 0.027	0.333 ± 0.007	0.717 ± 0.029	0.661 ± 0.027	0.983 ± 0.005	0.998 ± 0.001
Alea.	Seperate	0.508 ± 0.097	0.385 ± 0.052	0.739 ± 0.071	0.661 ± 0.066	0.969 ± 0.027	0.995 ± 0.005
Epis.	Joint	0.877 ± 0.032	0.854 ± 0.027	0.978 ± 0.010	0.984 ± 0.007	0.898 ± 0.054	0.991 ± 0.005
Epis.	Seperate	0.771 ± 0.051	0.707 ± 0.073	0.972 ± 0.010	0.978 ± 0.009	0.844 ± 0.049	0.986 ± 0.005

Table 6: OOD detection performance (Joint vs. separate branches).

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameters	Predicted Model	CUPID
Epoch	50	50
Batch size	256	256
Learning rate	0.001	0.0001
$\lambda_{1}$	/	0.001
$\lambda_{2}$	/	0.01

Appendix

Table 7: Hyperparameters for tabular examples.

Method	CUPID Alea.	CUPID Epis.	MC Dropout
AUC( $\uparrow$ )	0.965 ± 0.000	-	-
Accuracy( $\uparrow$ )	0.837 ± 0.000	-	-
AUC ( $\uparrow$ )	0.769 ± 0.006	0.688 ± 0.005	0.563 ± 0.001
AURC ( $\downarrow$ )	0.060 ± 0.001	0.088 ± 0.002	0.138 ± 0.000
Spearman ( $\uparrow$ )	0.812 ± 0.017	0.627 ± 0.012	0.365 ± 0.001

Appendix

Table 8: Performance of tabular examples.

Hyperparameters	Predicted Model	CUPID
Max epoch	50	50
Batch size	16	8
Learning rate	0.001	0.001
$\lambda_{1}$	/	0.001
$\lambda_{2}$	/	0.01

Appendix

Table 9: Hyperparameters for toy examples.

Figure 5: Data samples from GLV2 and HAM10000.

Dataset	Noise ( $\uparrow$ )	SNR ( $\downarrow$ )	Classes	Class balanced
GLV2	0.024	5.44 dB	2	True
HAM10000	0.022	12.78 dB	7	False

Appendix

Table 10: Dataset characteristics indicating different sources of uncertainty.

Figure 6: The structure of CUPID used in ResNet18 for the classification problem.

Hyperparameters	GLV2		HAM10000
Hyperparameters	Predicted	CUPID	Predicted	CUPID
Max epoch	10	15	10	15
Batch size	4	4	8	4
Learning rate	0.00001	0.00001	0.0001	0.00001
Decay step size	1	1	4.5	1
Decay $\gamma$	0.8	0.9	0.75	0.9

Appendix

Table 11: Hyperparameters for medical image classification experiments.

Method	GLV2		HAM10000
Method	AUC( $\uparrow$ )	Acc( $\uparrow$ )	AUC( $\uparrow$ )	Acc( $\uparrow$ )
CUPID/IGRUE/Rate-in	0.970 $\pm$ 0.000	0.909 $\pm$ 0.000	0.952 $\pm$ 0.000	0.821 $\pm$ 0.000
MC Dropout	0.979 $\pm$ 0.000	0.921 $\pm$ 0.000	0.946 $\pm$ 0.000	0.765 $\pm$ 0.002
PostNet	0.789 ± 0.004	0.718 ± 0.010	0.865 $\pm$ 0.002	0.664 $\pm$ 0.008
BNN	0.966 $\pm$ 0.005	0.914 $\pm$ 0.004	0.929 $\pm$ 0.004	0.742 $\pm$ 0.011
DEC	0.875 $\pm$ 0.002	0.878 $\pm$ 0.003	0.922 $\pm$ 0.014	0.768 $\pm$ 0.009

Appendix

Table 12: Mean accuracy and AUC for each classification model on GLV2 and HAM10000 datasets.

Figure 7: Data samples of ID and OOD datasets.

Hyperparameters	Predicted Model	CUPID
Max iteration	300000	300000
Batch size	16	16
Learning rate	0.0001	0.00001
$\lambda_{1}$	/	0.0001
$\lambda_{2}$	/	0.01

Appendix

Table 13: Hyperparameters for super-resolution.

Dataset	SSIM( $\uparrow$ )	PSNR( $\uparrow$ )
Set5	0.828	28.533
Set14	0.684	24.865
BSDS100	0.631	23.539
IXI	0.806	33.142

Appendix

Table 14: PSNR and SSIM for ESRGAN model.

Class Position	GLV2			HAM10000
Class Position	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
Aleatoric (4)	0.870 ± 0.002	0.018 ± 0.001	0.941 ± 0.004	0.769 ± 0.023	0.067 ± 0.007	0.722 ± 0.014
Aleatoric (3)	0.843 ± 0.008	0.022 ± 0.001	0.851 ± 0.006	0.751 ± 0.004	0.072 ± 0.003	0.624 ± 0.024
Aleatoric (2)	0.805 ± 0.014	0.026 ± 0.001	0.772 ± 0.019	0.749 ± 0.011	0.103 ± 0.006	0.675 ± 0.021
Epistemic (4)	0.769 ± 0.015	0.034 ± 0.002	0.701 ± 0.051	0.855 ± 0.006	0.047 ± 0.001	0.907 ± 0.001
Epistemic (3)	0.786 ± 0.003	0.033 ± 0.004	0.696 ± 0.015	0.869 ± 0.005	0.045 ± 0.000	0.898 ± 0.004

Appendix

Table 15: Results of inserting CUPID at different intermediate layers. Best-performing results for each metric are highlighted in bold.

Method	GLV2		PAPILA	ACRIMA	CIFAR10
Method	rAULC ( $\uparrow$ )	UCE ( $\downarrow$ )	OOD-UCE ( $\downarrow$ )
CUPID Alea.	0.840 ± 0.003	0.042 ± 0.004	0.308 ± 0.026	0.226 ± 0.079	0.273 ± 0.004
CUPID Epis.	0.649 ± 0.031	0.033 ± 0.008	0.115 ± 0.016	0.240 ± 0.006	0.633 ± 0.017
MC Dropout	0.682 ± 0.009	0.052 ± 0.019	0.257 ± 0.001	0.307 ± 0.045	0.699 ± 0.021
Rate-in	0.762 ± 0.011	0.038 ± 0.009	0.309 ± 0.009	0.411 ± 0.008	0.826 ± 0.019
IGRUE	0.375 ± 0.024	0.145 ± 0.012	0.154 ± 0.046	0.252 ± 0.016	0.596 ± 0.010

Appendix

Table 16: Uncertainty calibration results for misclassification (GLV2) and OOD detection (PAPILA, ACRIMA, CIFAR-10)

Method		Aleatoric			Epistemic
Method		AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
Block	12	0.725 ± 0.028	0.098 ± 0.010	0.614 ± 0.027	0.890 ± 0.003	0.052 ± 0.001	0.889 ± 0.004
	14	0.725 ± 0.013	0.107 ± 0.003	0.604 ± 0.035	0.902 ± 0.004	0.055 ± 0.003	0.886 ± 0.001
	16	0.749 ± 0.011	0.103 ± 0.006	0.675 ± 0.021	0.901 ± 0.004	0.058 ± 0.001	0.888 ± 0.005
	18	0.735 ± 0.024	0.100 ± 0.010	0.628 ± 0.049	0.895 ± 0.004	0.054 ± 0.002	0.886 ± 0.001

Appendix

Table 17: Performance of trunk block depth variants on CUPID for the classification task on HAM10000 dataset. Best-performing results for each metric are highlighted in bold.

Loss	Aleatoric			Epistemic
Loss	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
Max	0.749 ± 0.011	0.103 ± 0.006	0.675 ± 0.021	0.901 ± 0.004	0.058 ± 0.001	0.888 ± 0.005
No max	0.748 ± 0.005	0.098 ± 0.006	0.671 ± 0.025	0.898 ± 0.002	0.056 ± 0.003	0.888 ± 0.002

Appendix

Table 18: Performance of loss function on CUPID for the classification task on HAM10000 dataset. Best-performing results for each metric are highlighted in bold.

Method		Aleatoric			Epistemic
Method		AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )	AUC ( $\uparrow$ )	AURC ( $\downarrow$ )	Spearman ( $\uparrow$ )
$\lambda_{1}$	0.02	0.868 ± 0.001	0.018 ± 0.000	0.929 ± 0.019	0.743 ± 0.044	0.045 ± 0.013	0.676 ± 0.021
	0.01	0.870 ± 0.002	0.018 ± 0.001	0.941 ± 0.004	0.769 ± 0.015	0.034 ± 0.002	0.701 ± 0.051
	0.001	0.868 ± 0.005	0.018 ± 0.001	0.936 ± 0.010	0.754 ± 0.014	0.042 ± 0.005	0.674 ± 0.031

Appendix

Table 19: Performance of

\lambda_{1}

choice on CUPID for the classification task on GLV2 dataset.

Method and $\lambda_{1}$		PAPILA		ACRIMA		CIFAR10
Method and $\lambda_{1}$		AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )	AUC( $\uparrow$ )	AUPR( $\uparrow$ )
Alea.	0.02	0.457 ± 0.087	0.362 ± 0.043	0.727 ± 0.124	0.651 ± 0.149	0.973 ± 0.017	0.996 ± 0.003
	0.01	0.379 ± 0.027	0.333 ± 0.007	0.717 ± 0.029	0.661 ± 0.027	0.983 ± 0.005	0.998 ± 0.001
	0.001	0.389 ± 0.030	0.342 ± 0.016	0.718 ± 0.046	0.669 ± 0.062	0.984 ± 0.006	0.999 ± 0.001
Epis.	0.02	0.802 ± 0.039	0.743 ± 0.058	0.904 ± 0.058	0.921 ± 0.046	0.528 ± 0.293	0.923 ± 0.063
	0.01	0.877 ± 0.032	0.854 ± 0.027	0.978 ± 0.010	0.984 ± 0.007	0.898 ± 0.054	0.991 ± 0.005

Appendix

Table 20: Performance of

\lambda_{1}

choice on CUPID for the OOD task.

Trunk Num		Set5			Set14
Trunk Num		Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )	Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )
Aleatoric	3	0.525 ± 0.002	0.010 ± 0.000	0.017 ± 0.007	0.527 ± 0.002	0.012 ± 0.000	0.049 ± 0.014
	4	0.528 ± 0.006	0.010 ± 0.000	0.045 ± 0.018	0.527 ± 0.002	0.012 ± 0.000	0.049 ± 0.005
	5	0.527 ± 0.007	0.010 ± 0.000	0.041 ± 0.019	0.527 ± 0.001	0.012 ± 0.000	0.047 ± 0.015
	6	0.528 ± 0.003	0.010 ± 0.000	0.045 ± 0.020	0.524 ± 0.003	0.012 ± 0.000	0.039 ± 0.006
Epistemic	3	0.416 ± 0.004	0.020 ± 0.000	0.254 ± 0.024	0.428 ± 0.005	0.020 ± 0.000	0.217 ± 0.003

Appendix

Table 21: Performance of trunk block depth variants on CUPID for the super-resolution task. Best-performing results for each metric are highlighted in bold.

Loss		Set5			Set14
Loss		Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )	Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )
Aleatoric	Raw	0.528 ± 0.006	0.010 ± 0.000	0.045 ± 0.018	0.527 ± 0.002	0.012 ± 0.000	0.049 ± 0.005
Aleatoric	Add Lipis	0.512 ± 0.006	0.010 ± 0.001	0.031 ± 0.010	0.518 ± 0.009	0.013 ± 0.001	0.051 ± 0.015
Epistemic	Raw	0.416 ± 0.005	0.018 ± 0.001	0.266 ± 0.007	0.449 ± 0.005	0.019 ± 0.000	0.226 ± 0.003
Epistemic	Add Lipis	0.389 ± 0.007	0.019 ± 0.000	0.253 ± 0.035	0.399 ± 0.003	0.021 ± 0.000	0.230 ± 0.016

Appendix

Table 22: Performance of

\lambda_{1}

choice on CUPID for the super-resolution task. Best-performing results for each metric are highlighted in bold.

$\lambda_{1}$		Set5			Set14
$\lambda_{1}$		Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )	Pearson ( $\uparrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )
Aleatoric	0.001	0.527 ± 0.002	0.010 ± 0.000	0.034 ± 0.014	0.528 ± 0.001	0.012 ± 0.000	0.064 ± 0.016
	0.0001	0.528 ± 0.006	0.010 ± 0.000	0.045 ± 0.018	0.527 ± 0.002	0.012 ± 0.000	0.049 ± 0.005
	0.00001	0.529 ± 0.002	0.010 ± 0.000	0.037 ± 0.017	0.529 ± 0.002	0.012 ± 0.000	0.076 ± 0.011
Epistemic	0.001	0.423 ± 0.004	0.018 ± 0.000	0.264 ± 0.012	0.455 ± 0.010	0.018 ± 0.000	0.205 ± 0.009
	0.0001	0.416 ± 0.005	0.018 ± 0.001	0.266 ± 0.007	0.449 ± 0.005	0.019 ± 0.000	0.226 ± 0.003

Appendix

Table 23: Performance of

\lambda_{1}

choice on CUPID for the super-resolution task.

Uncertainty	Error	Set5		Set14
Uncertainty	Error	Spearman ( $\uparrow$ )	Pearson ( $\uparrow$ )	Spearman ( $\uparrow$ )	Pearson ( $\uparrow$ )
Aleatoric	$L_{1}$	0.643 ± 0.002	0.560 ± 0.001	0.607 ± 0.002	0.569 ± 0.001
Aleatoric	$L_{2}$	0.647 ± 0.002	0.487 ± 0.003	0.611 ± 0.002	0.514 ± 0.004
Epistemic	$L_{1}$	0.229 ± 0.007	0.258 ± 0.005	0.312 ± 0.006	0.327 ± 0.005
Epistemic	$L_{2}$	0.229 ± 0.008	0.209 ± 0.005	0.312 ± 0.006	0.254 ± 0.003

Appendix

Table 24: Effect of error map selection on Pearson and Spearman metrics.

Uncertainty	Error	Set5		Set14
Uncertainty	Error	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )	AUSE ( $\downarrow$ )	UCE ( $\downarrow$ )
Aleatoric	$L_{1}$	0.009 ± 0.000	0.034 ± 0.007	0.011 ± 0.000	0.017 ± 0.008
Aleatoric	$L_{2}$	0.002 ± 0.000	0.358 ± 0.004	0.002 ± 0.000	0.269 ± 0.021
Epistemic	$L_{1}$	0.026 ± 0.000	0.320 ± 0.004	0.023 ± 0.000	0.231 ± 0.005
Epistemic	$L_{2}$	0.005 ± 0.000	0.420 ± 0.001	0.004 ± 0.000	0.411 ± 0.001

Appendix

Table 25: Effect of error map selection on AUSE and UCE metrics.

Operation	Training Time (s)	Testing Time (s)
CUPID	4776.38	10.06
MC Dropout	/	49.57
Rate-in	2.85	48.46
IGRUE	5292.41	11.26
PostNet	49932.08	51.04
BNN	54974.84	217.06

Appendix

Table 26: Runtime comparison between training and testing phases in classification model uncertainty estimation.

Operation	Training Time (s)	Testing Time (s)
CUPID	26511.74	1.02
BayesCap	81158.45	0.79
in-rotate	/	2.03
in-noise	/	2.86
med-dropout	/	2.28
med-noise	/	2.47

Appendix

Table 27: Runtime comparison between training and testing phases in regression model uncertainty estimation.

Figure 8: More visual results of the uncertainty feature map of CUPID towards super-resolution model.

Figure 9: More visual results of the uncertainty feature map of all the methods.

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Apr 27, 2026

Credal Concept Bottleneck Models for Epistemic-Aleatoric Uncertainty Decomposition

Concept Bottleneck Models (CBMs) predict through human-interpretable concepts, but they typically output point concept probabilities that conflate epistemic uncertainty (reducible model underspecification) with aleatoric uncertainty (irreducible input ambiguity). This makes concept-level uncertainty hard to interpret and, more importantly, hard to act upon. We introduce CREDENCE (Credal Ensemble Concept Estimation), a CBM framework that decomposes concept uncertainty by construction. CREDENCE represents each concept as a credal prediction (a probability interval), derives epistemic uncertainty from disagreement across diverse concept heads, and estimates aleatoric uncertainty via a dedicated ambiguity output trained to match annotator disagreement when available. The resulting signals support prescriptive decisions: automate low-uncertainty cases, prioritize data collection for high-epistemic cases, route high-aleatoric cases to human review, and abstain when both are high. Across several tasks, we show that epistemic uncertainty is positively associated with prediction errors, whereas aleatoric uncertainty closely tracks annotator disagreement, providing guidance beyond error correlation. Our implementation is available at the following link: https://github.com/Tankiit/Credal_Sets/tree/ensemble-credal-cbm

Tanmoy Mukherjee, Thomas Bailleux, Pierre Marquis +1

Univ. Artois, CNRS, CRIL, France · Institut Universitaire de France

Uncertainty Quantification Concept Bottleneck Models Aleatoric Uncertainty Decision-Making under Uncertainty Uncertainty Decomposition Epistemic Uncertainty

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Abstract

Uncertainty Quantification

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