A Novel Bayes' Theorem for Upper Probabilities
Michele Caprio Yusuf Sale Eyke Hüllermeier Insup Lee
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.