Asymptotic equivalence of empirical likelihood and Bayesian MAP

TL;DR

This paper shows that empirical likelihood (EL) is asymptotically equivalent to Bayesian MAP estimators, providing a probabilistic interpretation for EL and analyzing its consistency under misspecification.

Contribution

It demonstrates that EL can be viewed as an asymptotic form of Bayesian MAP, establishing a probabilistic foundation for EL and comparing it with other empirical discrepancies.

Findings

EL and Bayesian MAP are asymptotically equivalent under misspecification.

EL has a well-defined probabilistic interpretation in a Bayesian context.

Other empirical discrepancy-based estimators are generally inconsistent under misspecification.

Abstract

In this paper we are interested in empirical likelihood (EL) as a method of estimation, and we address the following two problems: (1) selecting among various empirical discrepancies in an EL framework and (2) demonstrating that EL has a well-defined probabilistic interpretation that would justify its use in a Bayesian context. Using the large deviations approach, a Bayesian law of large numbers is developed that implies that EL and the Bayesian maximum a posteriori probability (MAP) estimators are consistent under misspecification and that EL can be viewed as an asymptotic form of MAP. Estimators based on other empirical discrepancies are, in general, inconsistent under misspecification.