  Bayesian inference is attractive for its coherence and good frequentist
properties. However, it is a common experience that eliciting a honest prior
may be difficult and, in practice, people often take an {\em empirical Bayes}
approach, plugging empirical estimates of the prior hyperparameters into the
posterior distribution. Even if not rigorously justified, the underlying idea
is that, when the sample size is large, empirical Bayes leads to "similar"
inferential answers. Yet, precise mathematical results seem to be missing. In
this work, we give a more rigorous justification in terms of merging of Bayes
and empirical Bayes posterior distributions. We consider two notions of
merging: Bayesian weak merging and frequentist merging in total variation.
Since weak merging is related to consistency, we provide sufficient conditions
for consistency of empirical Bayes posteriors. Also, we show that, under
regularity conditions, the empirical Bayes procedure asymptotically selects the
value of the hyperparameter for which the prior mostly favors the "truth".
Examples include empirical Bayes density estimation with Dirichlet process
mixtures.
