For finite parameter spaces, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it is Bayes. Various relaxations of this classical equivalence have been established for infinite parameter spaces, but these extensions are each subject to technical conditions that limit their applicability, especially to modern (semi and nonparametric) statistical problems. Using results in mathematical logic and nonstandard analysis, we extend this equivalence to arbitrary statistical decision problems: informally, we show that, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it has infinitesimal excess Bayes risk. In contrast to existing results, our equivalence holds in complete generality, that is, without regularity conditions or restrictions on the model or loss function. We also derive a nonstandard analogue of Blyth’s method that yields sufficient conditions for admissibility, and 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.
This research was made possible through an NSERC Discovery Grant, Connaught Award, and U.S. Air Force Office of Scientific Research Grant #FA9550-15-1-0074.
The authors owe a debt of gratitude to William Weiss for detailed suggestions. We thank Gintarė Džiugaitė, Cameron Freer, and H. Jerome Keisler for early discussions and insights, and thank Nate Ackerman, Robert Anderson, Michael Evans, Jeffrey Negrea, Arno Pauly, and Aaron Smith for feedback on drafts and helpful discussions. We thank the anonymous referees for suggesting reformulations of Theorems 2.2, 5.18 and 6.6, and for the idea of pursuing the results in Section 7 and Appendix J. Finally, the authors would like to thank Peter Hoff for his course notes, which served as our first introduction to the topic. This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley.
"On extended admissible procedures and their nonstandard Bayes risk." Ann. Statist. 49 (4) 2053 - 2078, August 2021. https://doi.org/10.1214/20-AOS2026