Credal Machine Learning for Risk-Averse Decision Making

Timo Löhr; Paul Hofman; Maximilian Muschalik; Eyke Hüllermeier

cs.LGOct 8, 2026

Credal Machine Learning for Risk-Averse Decision Making

Authors: Timo Löhr, Paul Hofman, Maximilian Muschalik, Eyke Hüllermeier

Organizations: LMU Munich, MCML · LMU Munich, MCML, DFKI

Abstract

In many machine learning applications, it is necessary to guard against worst-case scenarios and predictions that could result in substantial losses. In principle, this can be achieved by training risk-averse predictive models that minimize loss functions such as conditional value-at-risk (CVaR), rather than relying on models that perform well on average. In practice, however, the effectiveness of this approach to risk aversion is undermined by the learner's uncertainty regarding the true loss distribution and, consequently, the true CVaR. To achieve reliable risk-aversion, we propose a method in which this (epistemic) uncertainty is represented in terms of credal sets, i.e., sets of probability distributions. More specifically, we develop an efficient yet reliable learner that produces predictions in the form of credal sets and combine it with a novel decision rule that maps each credal set to a single predictive distribution for CVaR minimization. Across classification, under distribution shift, and in reinforcement learning, our approach reliably avoids catastrophic decisions, while sacrificing little in expected performance.

Figures & tables

Figure 1: Risk-averse decision making. Exemplary medical setting where our method shows lower conditional value-at-risk and avoids catastrophic decisions.

Figure 2: Distribution shift on BloodMNIST. Set size and accuracy against shift severity, mean and standard deviation over three seeds. CreWare’s sets grow as accuracy declines, while the baselines’ barely grow.

	Near-OOD		Far-OOD
Method	CIFAR-100	T.ImageNet	SVHN	Places365
$\text{CreEns}_{0.0}$	$.917\scriptstyle\pm.000$	$.911\scriptstyle\pm.001$	$.970\scriptstyle\pm.003$	$.919\scriptstyle\pm.003$
CreBNN	$.867\scriptstyle\pm.007$	$.866\scriptstyle\pm.006$	$.892\scriptstyle\pm.016$	$.872\scriptstyle\pm.006$
CreWra	$.918\scriptstyle\pm.000$	$.912\scriptstyle\pm.001$	$.971\scriptstyle\pm.003$	$.920\scriptstyle\pm.003$
CreDro	$.918\scriptstyle\pm.001$	$.911\scriptstyle\pm.001$	$.972\scriptstyle\pm.001$	$.920\scriptstyle\pm.003$
$\text{CreRL}_{0.95}$	$.913\scriptstyle\pm.003$	$.908\scriptstyle\pm.001$	$.969\scriptstyle\pm.007$	$.917\scriptstyle\pm.003$

Table 1: Out-of-distribution detection with CIFAR-10 as iD data. AUROC, mean and standard deviation over three runs, best in bold . CreWare (ours) outperforms all baselines on near- and far-OoD data.

Figure 3: Risk-averse prediction under distribution shift with PathMNIST. Our method beats baselines in terms of CVaR, while sacrificing little in mean loss.

Figure 4: Offline driving task. Left : mean reward, CVaR reward, and crash rate vs. hazard cell observations $k$ . Right : routes at three values of $k$ (decreasing epistemic uncertainty). Credal $+$ rule (ours) acts safe, whereas the MLE always drives into the hazard and MLE $+$ rule does so while the hazard is under-observed.

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: AUROC, fitting time, and inference time (in seconds) of CreWare on CIFAR-10 as a function of the number of landmarks

M\in\{45000,20000,10000,5000,2000,1000\}

. Mean and standard deviation over three seeds are shown.

	Resources
GPU	NVIDIA A100-SXM4, 40 GB HBM2
CPU	2 $\times$ AMD EPYC 7742 (64 cores each)
System RAM	1.0 TB
OS / driver	Ubuntu (Linux 6.8), CUDA driver 580

Appendix

Table 2: Compute node used for the experiments. Jobs were scheduled with slurm and ran inside a Singularity container (PyTorch NGC 26.03). Each job was allocated a single GPU.

Dataset	Reference	License
CIFAR-10	( Krizhevsky, 2009 )	No formal license
CIFAR-10-C	( Hendrycks and Dietterich, 2019 )	Creative Commons Attribution 4.0 International
CIFAR-10-H	( Peterson et al., 2019 )	Creative Commons Attribution 4.0 International
MedMNIST	( Yang et al., 2021 )	Creative Commons Attribution 4.0 International
MedMNIST-C	( Salvo et al., 2024 )	Creative Commons Attribution 4.0 International
CIFAR-100	( Krizhevsky, 2009 )	No formal license

Appendix

Table 3: Dataset Licenses.

Dataset	Epochs	Batch Size	Val. Split	Optim.	LR	Momentum	Weight Dec.	Scheduler
CIFAR-10	200	128	0.1	SGD	0.1	0.9	5e-4	Cosine
BloodMNIST	100	128	0.1	Adam	0.001	0.0	0.0	Factor 0.1 at 50, 75
PathMNIST	100	128	0.1	Adam	0.001	0.0	0.0	Factor 0.1 at 50, 75

Appendix

Table 4: Hyperparameters for ResNet18.

	Send home	Treat flu	ICU
Healthy	$0$	$6$	$8$
Flu	$4$	$1$	$4$
Sepsis	$20$	$14$	$2$

Appendix

Table 5: Cost matrix of the triage problem (rows: true condition, columns: action).

				Share of		Used in
Condition	Presentation	Mean	Std.	condition	all patients	training
Healthy	Typical	$(0,0)$	$1.0$	$85\%$	$46.75\%$	$100\%$
	Atypical (subgroup)	$(-6,4)$	$0.8$	$15\%$	$8.25\%$	$100\%$
Flu	Typical	$(3,0)$	$1.0$	$100\%$	$25.00\%$	$100\%$
Sepsis	Typical	$(3,3)$	$0.8$	$60\%$	$12.00\%$	$100\%$
	Masked (resembles healthy)	$(-1.8,1.8)$	$0.5$	$25\%$	$5.00\%$	$100\%$

Appendix

Table 6: Components of the data-generating process (informative coordinates). Shares refer to the natural frequencies of the validation and test data. Within each condition they sum to

100\%

, as do the shares of all patients. The last column gives the fraction of each component that is used in training.

Figure 6: Distribution shift on CIFAR-10. The mean and standard deviation of the set size and accuracy are reported. The sets of CreWare increase accordingly with the diminishing accuracy, while the other sets increase less in size.

Figure 7: Coverage versus efficiency for CIFAR-10-H. The CreWare method performs competitively, staying in the high coverage region while having a good efficiency.

	Near-OOD		Far-OOD
Method	CIFAR-100	TinyImageNet	MNIST	SVHN	Textures	Places365
$\text{CreEns}_{0.0}$	$0.917\scriptstyle\pm 0.000$	$0.911\scriptstyle\pm 0.001$	$0.961\scriptstyle\pm 0.002$	$0.970\scriptstyle\pm 0.003$	$0.948\scriptstyle\pm 0.000$	$0.919\scriptstyle\pm 0.003$
CreBNN	$0.867\scriptstyle\pm 0.007$	$0.866\scriptstyle\pm 0.006$	$0.912\scriptstyle\pm 0.024$	$0.892\scriptstyle\pm 0.016$	$0.894\scriptstyle\pm 0.011$	$0.872\scriptstyle\pm 0.006$
CreWra	$0.918\scriptstyle\pm 0.000$	$0.912\scriptstyle\pm 0.001$	$0.961\scriptstyle\pm 0.002$	$0.971\scriptstyle\pm 0.003$	$0.950\scriptstyle\pm 0.001$	$0.920\scriptstyle\pm 0.003$
CreDro	$0.918\scriptstyle\pm 0.001$	$0.911\scriptstyle\pm 0.001$	$0.965\scriptstyle\pm 0.003$	$0.972\scriptstyle\pm 0.001$	$0.950\scriptstyle\pm 0.001$	$0.920\scriptstyle\pm 0.003$
$\text{CreRL}_{0.95}$	$0.913\scriptstyle\pm 0.003$	$0.908\scriptstyle\pm 0.001$	$0.953\scriptstyle\pm 0.013$	$0.969\scriptstyle\pm 0.007$	$0.944\scriptstyle\pm 0.001$	$0.917\scriptstyle\pm 0.003$

Appendix

Table 7: Out-of-distribution detection with CIFAR-10 on all near-OoD and far-OoD datasets. The mean and standard deviation of the AUROC over three runs are reported. The best method is shown in bold . Subscripts denote the method-specific parameter

\alpha

controlling the set size.

	Near-OOD		Far-OOD
Method	CIFAR-100	TinyImageNet	MNIST	SVHN	Textures	Places365
$\text{CreEns}_{0.0}$	$0.911\scriptstyle\pm 0.001$	$0.903\scriptstyle\pm 0.001$	$0.946\scriptstyle\pm 0.002$	$0.955\scriptstyle\pm 0.004$	$0.940\scriptstyle\pm 0.002$	$0.909\scriptstyle\pm 0.003$
CreBNN	$0.872\scriptstyle\pm 0.005$	$0.869\scriptstyle\pm 0.004$	$0.905\scriptstyle\pm 0.010$	$0.907\scriptstyle\pm 0.011$	$0.901\scriptstyle\pm 0.010$	$0.872\scriptstyle\pm 0.005$
CreWra	$0.911\scriptstyle\pm 0.001$	$0.903\scriptstyle\pm 0.001$	$0.946\scriptstyle\pm 0.002$	$0.955\scriptstyle\pm 0.004$	$0.940\scriptstyle\pm 0.002$	$0.909\scriptstyle\pm 0.003$
CreDro	$0.911\scriptstyle\pm 0.001$	$0.903\scriptstyle\pm 0.001$	$0.949\scriptstyle\pm 0.002$	$0.960\scriptstyle\pm 0.004$	$0.941\scriptstyle\pm 0.002$	$0.909\scriptstyle\pm 0.003$
$\text{CreRL}_{0.95}$	$0.909\scriptstyle\pm 0.003$	$0.901\scriptstyle\pm 0.000$	$0.935\scriptstyle\pm 0.018$	$0.960\scriptstyle\pm 0.008$	$0.934\scriptstyle\pm 0.003$	$0.905\scriptstyle\pm 0.002$

Appendix

Table 8: Out-of-distribution detection with CIFAR-10 using the set size as discriminator. The mean and standard deviation of the AUROC over three runs are reported. The best method is shown in bold . CreWare outperforms other methods across near-OoD and far-OoD datasets.

	Near-OOD		Far-OOD
Method	PathMNIST	TissueMNIST	DermaMNIST	RetinaMNIST	OCTMNIST	BreastMNIST
$\text{CreEns}_{0.0}$	$0.780\scriptstyle\pm 0.011$	$0.497\scriptstyle\pm 0.008$	$0.681\scriptstyle\pm 0.016$	$0.089\scriptstyle\pm 0.009$	$0.219\scriptstyle\pm 0.014$	$0.043\scriptstyle\pm 0.001$
CreBNN	$0.744\scriptstyle\pm 0.043$	$0.009\scriptstyle\pm 0.015$	$0.788\scriptstyle\pm 0.015$	$0.337\scriptstyle\pm 0.143$	$0.000\scriptstyle\pm 0.001$	$0.001\scriptstyle\pm 0.002$
CreWra	$0.773\scriptstyle\pm 0.015$	$0.440\scriptstyle\pm 0.100$	$0.637\scriptstyle\pm 0.023$	$0.067\scriptstyle\pm 0.003$	$0.180\scriptstyle\pm 0.045$	$0.025\scriptstyle\pm 0.001$
CreDro	$0.761\scriptstyle\pm 0.022$	$0.362\scriptstyle\pm 0.065$	$0.603\scriptstyle\pm 0.047$	$0.062\scriptstyle\pm 0.002$	$0.143\scriptstyle\pm 0.014$	$0.024\scriptstyle\pm 0.000$
$\text{CreRL}_{0.95}$	$0.573\scriptstyle\pm 0.021$	$0.500\scriptstyle\pm 0.000$	$0.288\scriptstyle\pm 0.006$	$0.105\scriptstyle\pm 0.000$	$0.226\scriptstyle\pm 0.000$	$0.043\scriptstyle\pm 0.000$

Appendix

Table 9: Out-of-distribution detection with BloodMNIST. The mean and standard deviation of the AUROC over three runs are reported. The best method is shown in bold . CreWare outperforms other methods across near-OoD and far-OoD datasets.

Figure 8: Train and test time in seconds (log-scale) on the full train and test splits of CIFAR-10. The mean and standard deviation over three runs are reported.

Figure 9: Risk-averse prediction under distribution shift with PathMNIST evaluated by log-loss.

Figure 10: Risk-averse prediction under distribution shift with BloodMNIST evaluated by log-loss.

Figure 11: Risk-averse prediction under distribution shift with PathMNIST evaluated by Brier-loss.

Figure 12: Risk-averse prediction under distribution shift with BloodMNIST evaluated by Brier-loss.

Figure 13: Risk-averse prediction under distribution shift with PathMNIST using different

\alpha

-values.

Figure 14: Risk-averse prediction under distribution shift with BloodMNIST using different

\alpha

-values.

Figure 15: Cost-sensitive decision making. CreWare shows lower conditional value-at-risk and avoids catastrophic decisions.

Figure 16: Ablation over the training coverage of the underrepresented subgroup (

\tau=0.1

10

seeds): our approach is nearly invariant to the missing data, while SQwash’s protection collapses at

0\%

. Shaded bands show the standard deviation of the mean over ten seeds.

Panel of Figure 17 — Figure 17: The comparison at high ( $6$ subgroup sepsis cases, a) and lower ( $30$ cases, b) epistemic uncertainty: the credal advantage narrows as evidence accumulates.

Figure 18: Offline driving task on the larger map: mean reward, CVaR reward, and crash rate vs. the number of hazard observations $k$ (left), with route snapshots at three values of $k$ (right) representing levels of epistemic uncertainty. Credal $+$ rule (ours) is the only agent safe at every uncertainty level (robust to unobserved danger via the credal set and to known tail risk via CVaR), while MLE always takes the shortcut and the point-estimate risk-averse ablation MLE $+$ rule drives into the hazard while it is under-observed.

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