METRON DOI 10.1007/s40300-014-0044-1 Empirical Bayes methods in classical and Bayesian inference Sonia Petrone · Stefano Rizzelli · Judith Rousseau · Catia Scricciolo Received: 27 April 2014 / Accepted: 5 May 2014 © Sapienza Università di Roma 2014 Abstract 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 empiri- cal 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. Keywords Bayesian weak merging · Compound experiments · Frequentist strong merging · Hyper-parameter oracle value · Latent distribution · Maximum marginal likelihood estimation · Shrinkage estimation 1 Introduction Empirical Bayes methods are popularly employed by researchers and practitioners and are attractive in appearing to bridge frequentist and Bayesian approaches to inference. In fact, a frequentist statistician would find just a formal Bayesian flavor in empirical Bayes methods, while a Bayesian statistician would say that there is nobody less Bayesian than an empirical Bayesian (Lindley, in [6]). Further confusing, in the literature the term empirical Bayes is used in quite diverse contexts, with different motivations. Classical empirical Bayes methods arose in the context of compound experiments, where a latent distribution driving experiment- specific parameters formally acts as a prior on each one such parameter and is estimated from the data, usually by maximum likelihood. The term empirical Bayes is also used in the context of purely Bayesian inference when hyper-parameters of a subjective prior distribu- S. Petrone (B ) · S. Rizzelli · C. Scricciolo Bocconi University, Milan, Italy e-mail: sonia.petrone@unibocconi.it J. Rousseau CREST-ENSAE and CEREMADE, Université Paris Dauphine, Paris, France 123