Dan Roy (University of Toronto)
Nonstandard complete class theorems

For finite parameter spaces under finite loss, there is a close link between optimal frequentist decision procedures and Bayesian procedures:

every Bayesian procedure derived from a prior with full support is admissible, and every admissible procedure is Bayes. This relationship breaks down as we move beyond finite parameter spaces. There is a long line of work relating admissible procedures to Bayesian ones in more general settings. Under some regularity conditions, admissible procedures can be shown to be the limit of Bayesian procedures. Under additional regularity, they are generalized Bayesian, i.e., they minimize the average loss 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 notion of a hyperfinite statistical decision problem defined on a hyperfinite probability space and study the class of nonstandard Bayesian decision procedures---namely, those whose average risk with respect to some prior is within an infinitesimal of the optimal Bayes risk. We show that if there is a suitable hyperfinite approximation to a standard statistical decision problem, then every admissible decision procedure is nonstandard Bayes, and so the nonstandard Bayesian procedures form a complete class. We give sufficient regularity conditions on standard statistical decision problems admitting hyperfinite approximations. Joint work with Haosui (Kevin) Duanmu.