# A Novel Bayes' Theorem for Upper Probabilities

arXiv: https://arxiv.org/abs/2307.06831 · HTML: https://mlsanity.com/paper/2307.06831/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 33 references that are in the map.
- First posted on arXiv 13 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [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)
- [Understanding Measures of Uncertainty for Adversarial Example Detection](https://mlsanity.com/paper/1803.08533/) (2018)
- [Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?](https://mlsanity.com/paper/2209.03302/) (2022)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification](https://mlsanity.com/paper/2203.16481/) (2022)
- [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)
- [Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox](https://mlsanity.com/paper/1712.08946/) (2017)

## Built on by (most cited first, 13 of 13)

- [Second-Order Uncertainty Quantification: A Distance-Based Approach](https://mlsanity.com/paper/2312.00995/) (2023)
- [Credal Learning Theory](https://mlsanity.com/paper/2402.00957/) (2024)
- [From Risk to Uncertainty: Generating Predictive Uncertainty Measures via Bayesian Estimation](https://mlsanity.com/paper/2402.10727/) (2024)
- [Conformalized Credal Regions for Classification with Ambiguous Ground Truth](https://mlsanity.com/paper/2411.04852/) (2024)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Statistical Multicriteria Benchmarking via the GSD-Front](https://mlsanity.com/paper/2406.03924/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Uncertainty-based Offline Variational Bayesian Reinforcement Learning for Robustness under Diverse Data Corruptions](https://mlsanity.com/paper/2411.00465/) (2024)
- [Credal and Interval Deep Evidential Classifications](https://mlsanity.com/paper/2512.05526/) (2025)
- [Conformal Prediction Regions are Imprecise Highest Density Regions](https://mlsanity.com/paper/2502.06331/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Generalized Regularized Evidential Deep Learning Models: Theory and Comprehensive Evaluation](https://mlsanity.com/paper/2512.23753/) (2025)
- [Robust Predictive Uncertainty and Double Descent in Contaminated Bayesian Random Features](https://mlsanity.com/paper/2602.19126/) (2026)

## Abstract (arXiv)

In their seminal 1990 paper, Wasserman and Kadane establish an upper bound for the Bayes' posterior probability of a measurable set A, when the prior lies in a class of probability measures P and the likelihood is precise. They also give a sufficient condition for such upper bound to hold with equality. In this paper, we introduce a generalization of their result by additionally addressing uncertainty related to the likelihood. We give an upper bound for the posterior probability when both the prior and the likelihood belong to a set of probabilities. Furthermore, we give a sufficient condition for this upper bound to become an equality. This result is interesting on its own, and has the potential of being applied to various fields of engineering (e.g. model predictive control), machine learning, and artificial intelligence.