Date/Time
Date(s) - 25/10/2024
3:30 pm - 4:30 pm

Location: HH-305

Speaker: Daniel Roy (Canada CIFAR AI Chair, Research Director of the Vector Institute, UofT)

Title: Sets of Infinitesimal Measure: Closing Long-standing, Open Problems in Statistical Decision Theory via Nonstandard Analysis

Research Area: machine learning, statistics, applied probability; CIFAR Canada AI Chair; Director, Vector Institute.

Abstract: In this talk, I’ll summarize recent work exploiting tools in mathematical logic to resolve long-standing open problems in statistical decision theory. I’ll focus on an exact characterization of admissibility in terms of Bayes optimality in the nonstandard extension of the original decision problem, as introduced by Duanmu and Roy (Ann. Statist. 49(4): DOI:10.1214/20-AOS2026). Unlike the consideration of improper priors or other generalized notions of Bayes optimality, the nonstandard extension is distinguished, in part, by having priors that can assign “infinitesimal” mass in a sense that is made rigorous using results from nonstandard analysis. With these additional priors, we find that, informally speaking, a decision procedure $\delta_0$ is admissible in an ordinary statistical decision problem if and only if, in the problem’s “nonstandard extension”, the nonstandard extension of $\delta_0$ is Bayes optimal among the extensions of standard decision procedures with respect to a nonstandard prior assigning at least infinitesimal mass to every standard parameter value. We use this theorem to give further characterizations of admissibility, one related to Blyth’s method and another related to a condition due to Stein that characterizes admissibility under regularity. Our results imply that Blyth’s method is a sound and complete method for establishing admissibility. Buoyed by this result, we revisit the univariate two-sample common-mean problem and show that the Graybill–Deal estimator is admissible among a certain class of unbiased decision procedures.

** Joint work with Haosui Duanmu and David Schrittesser **

Coffee will be served in the same room – HH 305 at 3pm – all are welcome.