Admissibility is Bayes optimality with infinitesimals

12/28/2021
by   Haosui Duanmu, et al.
0

We give an exact characterization of admissibility in statistical decision problems in terms of Bayes optimality in a so-called nonstandard extension of the original decision problem, as introduced by Duanmu and Roy. Unlike the consideration of improper priors or other generalized notions of Bayes optimalitiy, the nonstandard extension is distinguished, in part, by having priors that can assign "infinitesimal" mass in a sense that can be made rigorous using results from nonstandard analysis. With these additional priors, we find that, informally speaking, a decision procedure δ_0 is admissible in the original statistical decision problem if and only if, in the nonstandard extension of the problem, the nonstandard extension of δ_0 is Bayes optimal among the extensions of standard decision procedures with respect to a nonstandard prior that assigns at least infinitesimal mass to every standard parameter value. We use the above theorem to give further characterizations of admissibility, one related to Blyth's method, one to a condition due to Stein which characterizes admissibility under some regularity assumptions; and finally, a characterization using finitely additive priors in decision problems meeting certain regularity requirements. 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/26/2020

Matrix superharmonic priors for Bayes estimation under matrix quadratic loss

We investigate Bayes estimation of a normal mean matrix under the matrix...
research
12/26/2022

Statistical minimax theorems via nonstandard analysis

For statistical decision problems with finite parameter space, it is wel...
research
07/02/2020

Posterior Model Adaptation With Updated Priors

Classification approaches based on the direct estimation and analysis of...
research
11/21/2017

Constrained empirical Bayes priors on regression coefficients

In the context of model uncertainty and selection, empirical Bayes proce...
research
08/18/2023

Minimaxity under half-Cauchy type priors

This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis),...
research
01/31/2023

Multicalibration as Boosting for Regression

We study the connection between multicalibration and boosting for square...
research
06/04/2022

A Further Look at the Bayes Blind Spot

Gyenis and Redei have demonstrated that any prior p on a finite algebra,...

Please sign up or login with your details

Forgot password? Click here to reset