Bayes’ theorem for Choquet capacities. (English) Zbl 0736.62026

Wasserman, Larry A.; Kadane, Joseph B.

Ann. Stat. 18, No. 3, 1328-1339 (1990).

Authors’ 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 \({\mathcal P}\). The bound is a rational function of two Choquet integrals. If \({\mathcal P}\) 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.

Reviewer: P.Ressel (Eichstätt)

MSC:

62F15	Bayesian inference
62B99	Sufficiency and information
62F35	Robustness and adaptive procedures (parametric inference)
62A01	Foundations and philosophical topics in statistics

Keywords:

posterior bounds; rational function of two Choquet integrals; weakly compact; majorization; priors; robust Bayesian inference; characterization of 2-alternating Choquet capacities