A Novel Bayes' Theorem for Upper Probabilities

TL;DR

This paper generalizes Wasserman and Kadane's 1990 upper bound for Bayesian posterior probabilities to include uncertainty in both priors and likelihoods, providing new bounds and conditions for equality.

Contribution

It introduces a generalized upper bound for posterior probabilities considering uncertainty in prior and likelihood, extending previous results and applicable to AI and engineering.

Findings

Provides an upper bound for posterior probability with uncertain prior and likelihood

Establishes a sufficient condition for the upper bound to be tight

Potential applications in machine learning and control systems

Abstract

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 $\mathcal{P}$ 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.