Bayes and empirical Bayes : Do they merge?

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