Maximum a Posteriori Estimators as a Limit of Bayes Estimators

TL;DR

This paper challenges the common belief that MAP estimators are a limit of Bayes estimators with 0-1 loss, providing a counterexample and a corrected condition using variational analysis.

Contribution

It demonstrates that the usual claim about MAP being a limit of Bayes estimators is false in general and offers a corrected condition for when it holds.

Findings

Counterexample disproves the general claim

Provides a level-set condition for the claim to hold

Utilizes variational analysis tools in Bayesian estimation

Abstract

Maximum a posteriori and Bayes estimators are two common methods of point estimation in Bayesian Statistics. It is commonly accepted that maximum a posteriori estimators are a limiting case of Bayes estimators with 0-1 loss. In this paper, we provide a counterexample which shows that in general this claim is false. We then correct the claim that by providing a level-set condition for posterior densities such that the result holds. Since both estimators are defined in terms of optimization problems, the tools of variational analysis find a natural application to Bayesian point estimation.