# Credal Bayesian Deep Learning

arXiv: https://arxiv.org/abs/2302.09656 · HTML: https://mlsanity.com/paper/2302.09656/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 82 references that are in the map.
- First posted on arXiv 19 Feb 2023.

## Builds on (most cited first, 23 of 23)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [Conformal prediction beyond exchangeability](https://mlsanity.com/paper/2202.13415/) (2022)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Puzzle Mix: Exploiting Saliency and Local Statistics for Optimal Mixup](https://mlsanity.com/paper/2009.06962/) (2020)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Posterior Network: Uncertainty Estimation without OOD Samples via Density-Based Pseudo-Counts](https://mlsanity.com/paper/2006.09239/) (2020)
- [MNIST-C: A Robustness Benchmark for Computer Vision](https://mlsanity.com/paper/1906.02337/) (2019)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [PAC Confidence Sets for Deep Neural Networks via Calibrated Prediction](https://mlsanity.com/paper/2001.00106/) (2019)
- [Model Selection in Bayesian Neural Networks via Horseshoe Priors](https://mlsanity.com/paper/1705.10388/) (2017)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [AutoDEUQ: Automated Deep Ensemble with Uncertainty Quantification](https://mlsanity.com/paper/2110.13511/) (2021)
- [BNNpriors: A library for Bayesian neural network inference with different prior distributions](https://mlsanity.com/paper/2105.06964/) (2021)
- [Physics Constrained Motion Prediction with Uncertainty Quantification](https://mlsanity.com/paper/2302.01060/) (2023)
- [Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox](https://mlsanity.com/paper/1712.08946/) (2017)
- [An Evidential Neural Network Model for Regression Based on Random Fuzzy Numbers](https://mlsanity.com/paper/2208.00647/) (2022)

## Built on by (most cited first, 30 of 45)

- [Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?](https://mlsanity.com/paper/2306.09586/) (2023)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [A Novel Bayes' Theorem for Upper Probabilities](https://mlsanity.com/paper/2307.06831/) (2023)
- [Credal Wrapper of Model Averaging for Uncertainty Estimation in Classification](https://mlsanity.com/paper/2405.15047/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [A Unified Evaluation Framework for Epistemic Predictions](https://mlsanity.com/paper/2501.16912/) (2025)
- [Position: Epistemic uncertainty estimation methods are fundamentally incomplete](https://mlsanity.com/paper/2505.23506/) (2025)
- [Statistical Multicriteria Evaluation of LLM-Generated Text](https://mlsanity.com/paper/2506.18082/) (2025)
- [Uncertainty-based Offline Variational Bayesian Reinforcement Learning for Robustness under Diverse Data Corruptions](https://mlsanity.com/paper/2411.00465/) (2024)
- [A calibration test for evaluating set-based epistemic uncertainty representations](https://mlsanity.com/paper/2502.16299/) (2025)
- [Learning Credal Ensembles via Distributionally Robust Optimization](https://mlsanity.com/paper/2602.08470/) (2026)
- [IBCL: Zero-shot Model Generation under Stability-Plasticity Trade-offs](https://mlsanity.com/paper/2305.14782/) (2023)
- [Generalising realisability in statistical learning theory under epistemic uncertainty](https://mlsanity.com/paper/2402.14759/) (2024)
- [Generalization Bounds and Stopping Rules for Learning with Self-Selected Data](https://mlsanity.com/paper/2505.07367/) (2025)
- [Off-Switching Not Guaranteed](https://mlsanity.com/paper/2502.08864/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Bayesian Generalization Error in Linear Neural Networks with Concept Bottleneck Structure and Multitask Formulation](https://mlsanity.com/paper/2303.09154/) (2023)
- [Distilling Calibration via Conformalized Credal Inference](https://mlsanity.com/paper/2501.06066/) (2025)
- [Optimal Conformal Prediction under Epistemic Uncertainty](https://mlsanity.com/paper/2505.19033/) (2025)
- [HybridFlow: Quantification of Aleatoric and Epistemic Uncertainty with a Single Hybrid Model](https://mlsanity.com/paper/2510.05054/) (2025)
- [Towards Better Understanding of In-Context Learning Ability from In-Context Uncertainty Quantification](https://mlsanity.com/paper/2405.15115/) (2024)
- [Adaptive Conformal Guidance for Learning under Uncertainty](https://mlsanity.com/paper/2502.16736/) (2025)
- [Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference](https://mlsanity.com/paper/2505.19136/) (2025)
- [Representative, Informative, and De-Amplifying: Requirements for Robust Bayesian Active Learning under Model Misspecification](https://mlsanity.com/paper/2506.07805/) (2025)
- [Unreliable Uncertainty Estimates with Monte Carlo Dropout](https://mlsanity.com/paper/2512.14851/) (2025)
- [Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluation](https://mlsanity.com/paper/2512.23753/) (2025)
- [Bayesian Conformal Prediction as a Decision Risk Problem](https://mlsanity.com/paper/2602.03331/) (2026)
- [Quantification of Credal Uncertainty: A Distance-Based Approach](https://mlsanity.com/paper/2603.27270/) (2026)

## Abstract (arXiv)

Uncertainty quantification and robustness to distribution shifts are important goals in machine learning and artificial intelligence. Although Bayesian Neural Networks (BNNs) allow for uncertainty in the predictions to be assessed, different sources of predictive uncertainty cannot be distinguished properly. We present Credal Bayesian Deep Learning (CBDL). Heuristically, CBDL allows to train an (uncountably) infinite ensemble of BNNs, using only finitely many elements. This is possible thanks to prior and likelihood finitely generated credal sets (FGCSs), a concept from the imprecise probability literature. Intuitively, convex combinations of a finite collection of prior-likelihood pairs are able to represent infinitely many such pairs. After training, CBDL outputs a set of posteriors on the parameters of the neural network. At inference time, such posterior set is used to derive a set of predictive distributions that is in turn utilized to distinguish between (predictive) aleatoric and epistemic uncertainties, and to quantify them. The predictive set also produces either (i) a collection of outputs enjoying desirable probabilistic guarantees, or (ii) the single output that is deemed the best, that is, the one having the highest predictive lower probability -- another imprecise-probabilistic concept. CBDL is more robust than single BNNs to prior and likelihood misspecification, and to distribution shift. We show that CBDL is better at quantifying and disentangling different types of (predictive) uncertainties than single BNNs and ensemble of BNNs. In addition, we apply CBDL to two case studies to demonstrate its downstream tasks capabilities: one, for motion prediction in autonomous driving scenarios, and two, to model blood glucose and insulin dynamics for artificial pancreas control. We show that CBDL performs better when compared to an ensemble of BNNs baseline.