Empirical Bayes methods in classical and Bayesian inference

Empirical Bayes methods are often thought of as a bridge between classical and Bayesian inference. In fact, in the literature the term empirical Bayes is used in quite diverse contexts and with different motivations. In this article, we provide a brief overview of empirical Bayes methods highlighting their scopes and meanings in different problems. We focus on recent results about merging of Bayes and empirical Bayes posterior distributions that regard popular, but otherwise debatable, empirical Bayes procedures as computationally convenient approximations of Bayesian solutions.