Bayes' Theorem for Choquet Capacities

We give an upper bound for the posterior probability of a measurable set A when the prior lies in a class of probability measures 9. The bound is a rational function of two Choquet integrals. If g; 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. 1. Introduction. Sets of probability measures arise naturally in classical robustness, Bayesian robustness and group decision making. Some sets of probability measures give rise to upper probabilities that are 2-alternating Choquet capacities. These upper probabilities are pervasive in the robustness The purpose of this paper is to prove a version of Bayes' theorem for sets of prior probabilities that have the 2-alternating property and to see how these sets of probabilities may be exploited in Bayesian robustness. This result was proved in Wasserman (1988, 1990) for infinitely alternating capacities (also known as belief functions ; Shafer, 1976). Since infinitely alternating capacities are also 2-alternating , the present result generalizes that theorem. However, the proof in the infinitely alternating case uses an argument that depends on properties that infinitely alternating capacities possess that are not shared by 2-alternating capacities in general. Also, in this paper, the conditions given are both necessary and sufficient. A proof of sufficiency when the parameter space is finite is given in Walley (1981). Section 2 of this paper states and proves the main result. In Section 3, we apply the result to derive explicit bounds for the posterior probability of a measurable set using various classes of priors. Section 4 contains a discussion.