Bayes and empirical Bayes: do they merge?

Bayesian inference is attractive for its internal coherence and for often having good frequentist properties. However, eliciting a honest prior may be difficult and a common practice is to take an empirical Bayes approach using some estimate of the prior hyperparameters. Although not rigorous, the underlying idea is that, for a sufficiently large sample size, empirical Bayes methods should lead to similar inferential answers as a proper Bayesian inference. However, precise mathematical results on this asymptotic agreement seem to be missing. In this work, we give results in terms of merging of Bayesian and empirical Bayes posterior distributions. We study two notions of merging: Bayesian weak merging and frequentist merging in total variation. We also show that, under regularity conditions, the empirical Bayes approach asymptotically gives an oracle selection of the prior hyperparameters. Examples include empirical Bayes density estimation with Dirichlet process mixtures.