Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator

TL;DR

This paper analyzes the asymptotic behavior of empirical Bayes posteriors derived from maximum marginal likelihood estimators, providing bounds and demonstrating their effectiveness across various models.

Contribution

It characterizes the estimator's location, establishes contraction rate bounds, and compares empirical and hierarchical Bayes posteriors in a general setting.

Findings

Empirical Bayes posteriors achieve optimal contraction rates.

Hierarchical Bayes posteriors match empirical Bayes contraction rates.

Results apply to various models and prior distributions.

Abstract

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.