# Credal Learning Theory

arXiv: https://arxiv.org/abs/2402.00957 · HTML: https://mlsanity.com/paper/2402.00957/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 80 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Builds on (most cited first, 30 of 32)

- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [In Search of Lost Domain Generalization](https://mlsanity.com/paper/2007.01434/) (2020)
- [Generalizing to Unseen Domains: A Survey on Domain Generalization](https://mlsanity.com/paper/2103.03097/) (2021)
- [Domain Generalization: A Survey](https://mlsanity.com/paper/2103.02503/) (2021)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Domain Generalization via Invariant Feature Representation](https://mlsanity.com/paper/1301.2115/) (2013)
- [SWAD: Domain Generalization by Seeking Flat Minima](https://mlsanity.com/paper/2102.08604/) (2021)
- [Generalizing to unseen domains via distribution matching](https://mlsanity.com/paper/1911.00804/) (2019)
- [A survey on domain adaptation theory: learning bounds and theoretical guarantees](https://mlsanity.com/paper/2004.11829/) (2020)
- [Towards a Theoretical Framework of Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.04496/) (2021)
- [Domain Generalization via Multidomain Discriminant Analysis](https://mlsanity.com/paper/1907.11216/) (2019)
- [Credal Bayesian Deep Learning](https://mlsanity.com/paper/2302.09656/) (2023)
- [Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?](https://mlsanity.com/paper/2306.09586/) (2023)
- [Optimal Testing for Properties of Distributions](https://mlsanity.com/paper/1507.05952/) (2015)
- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Continual Self-supervised Learning: Towards Universal Multi-modal Medical Data Representation Learning](https://mlsanity.com/paper/2311.17597/) (2023)
- [An Online Learning Approach to Interpolation and Extrapolation in Domain Generalization](https://mlsanity.com/paper/2102.13128/) (2021)
- [Robust Hypothesis Testing Using Wasserstein Uncertainty Sets](https://mlsanity.com/paper/1805.10611/) (2018)
- [A Generalization Error Bound for Multi-class Domain Generalization](https://mlsanity.com/paper/1905.10392/) (2019)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Integral Probability Metrics PAC-Bayes Bounds](https://mlsanity.com/paper/2207.00614/) (2022)
- [Credal Self-Supervised Learning](https://mlsanity.com/paper/2106.11853/) (2021)
- [Domain Generalization -- A Causal Perspective](https://mlsanity.com/paper/2209.15177/) (2022)
- [A Novel Bayes' Theorem for Upper Probabilities](https://mlsanity.com/paper/2307.06831/) (2023)
- [Understanding Robust Learning through the Lens of Representation Similarities](https://mlsanity.com/paper/2206.09868/) (2022)
- [Just a Glimpse: Rethinking Temporal Information for Video Continual Learning](https://mlsanity.com/paper/2305.18418/) (2023)
- [Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox](https://mlsanity.com/paper/1712.08946/) (2017)
- [On the Calibration of Probabilistic Classifier Sets](https://mlsanity.com/paper/2205.10082/) (2022)
- [Valid inferential models for prediction in supervised learning problems](https://mlsanity.com/paper/2112.10234/) (2021)

## Built on by (most cited first, 30 of 31)

- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [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)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Conformal Prediction Regions are Imprecise Highest Density Regions](https://mlsanity.com/paper/2502.06331/) (2025)
- [Random-Set Large Language Models](https://mlsanity.com/paper/2504.18085/) (2025)
- [Truthful Elicitation of Imprecise Forecasts](https://mlsanity.com/paper/2503.16395/) (2025)
- [Generalising realisability in statistical learning theory under epistemic uncertainty](https://mlsanity.com/paper/2402.14759/) (2024)
- [Epistemic Wrapping for Uncertainty Quantification](https://mlsanity.com/paper/2505.02277/) (2025)
- [Generalization Bounds and Stopping Rules for Learning with Self-Selected Data](https://mlsanity.com/paper/2505.07367/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Optimal Conformal Prediction under Epistemic Uncertainty](https://mlsanity.com/paper/2505.19033/) (2025)
- [Random-Set Graph Neural Networks](https://mlsanity.com/paper/2605.11987/) (2026)
- [Towards Better Understanding of In-Context Learning Ability from In-Context Uncertainty Quantification](https://mlsanity.com/paper/2405.15115/) (2024)
- [Credal Graph Neural Networks](https://mlsanity.com/paper/2512.02722/) (2025)
- [Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluation](https://mlsanity.com/paper/2512.23753/) (2025)
- [Quantification of Credal Uncertainty: A Distance-Based Approach](https://mlsanity.com/paper/2603.27270/) (2026)
- [Credal Large Language Models for Semantic Commitment under Uncertainty](https://mlsanity.com/paper/2608.23244/) (2026)
- [Reliable Classification with Conformal Learning and Interval-Type 2 Fuzzy Sets](https://mlsanity.com/paper/2504.15360/) (2025)
- [Epistemic Errors of Imperfect Multitask Learners When Distributions Shift](https://mlsanity.com/paper/2505.23496/) (2025)
- [When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates](https://mlsanity.com/paper/2510.04769/) (2025)
- [Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference](https://mlsanity.com/paper/2511.21223/) (2025)
- [Efficient Credal Prediction through Decalibration](https://mlsanity.com/paper/2603.08495/) (2026)
- [Structured Credal Learning](https://mlsanity.com/paper/2603.14070/) (2026)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)
- [A formal definition and meta-model for a machine theory of mind](https://mlsanity.com/paper/2606.03471/) (2026)
- [SSP-RACL: Classification of Noisy Fundus Images with Self-Supervised Pretraining and Robust Adaptive Credal Loss](https://mlsanity.com/paper/2409.18147/) (2024)
- [Optimal Transport for ε-Contaminated Credal Sets: To the Memory of Sayan Mukherjee](https://mlsanity.com/paper/2410.03267/) (2024)
- [Upper Entropy for 2-Monotone Lower Probabilities](https://mlsanity.com/paper/2603.23558/) (2026)
- [The Physical Cutoff Does Not Restore Homogenization: Phase-Dependent Burning in the Strain G-Equation](https://mlsanity.com/paper/2608.15337/) (2026)

## Abstract (arXiv)

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.