Bayes' Theorem for Choquet Capacities

Abstract

We give an upper bound for the posterior probability of a measurable set $A$ when the prior lies in a class of probability measures $\mathscr{P}$. The bound is a rational function of two Choquet integrals. If $\mathscr{P}$ is weakly compact and is closed with respect to majorization, then the bound is sharp if and only if the upper prior probability is 2-alternating. The result is used to compute bounds for several sets of priors used in robust Bayesian inference. The result may be regarded as a characterization of 2-alternating Choquet capacities.