On Extended Admissible Procedures and their Nonstandard Bayes Risk

Abstract

For finite parameter spaces under finite loss, every Bayes procedure derived from a prior with full support is admissible, and every admissible procedure is Bayes. This relationship already breaks down once we move to finite-dimensional Euclidean parameter spaces. Compactness and strong regularity conditions suffice to repair the relationship, but without these conditions, admissible procedures need not be Bayes. Under strong regularity conditions, admissible procedures can be shown to be the limits of Bayes procedures. Under even stricter conditions, they are generalized Bayes, i.e., they minimize the Bayes risk with respect to an improper prior. In both these cases, one must venture beyond the strict confines of Bayesian analysis. Using methods from mathematical logic and nonstandard analysis, we introduce the class of nonstandard Bayes decision procedures---namely, those whose Bayes risk with respect to some prior is within an infinitesimal of the optimal Bayes risk. Among procedures with finite risk functions, we show that a decision procedure is extended admissible if and only if its nonstandard extension is nonstandard Bayes. For problems with continuous risk functions defined on metric parameter spaces, we derive a nonstandard analogue of Blyth's method that can be used to establish the admissibility of a procedure. We also apply the nonstandard theory to derive a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.