Seminar
Stefano Rizzelli
Università Cattolica del Sacro Cuore - École Polytecnique Fédérale de Lausanne

The Bayesian inferential paradigm prescribes the specification of a prior distribution on the parameters of the statistical model. For complex models, the subjective elicitation of prior hyper-parameters can be a delicate and difficult task. This is particularly the case for hyper-parameters affecting posterior inference via complexity penalization, shrinkage effects, etc. In absence of sufficient information a priori, a principled specification of a hyper-prior distribution can be difficult too and complicate computations. It is common practice to resort to a data-driven choice of the prior hyper-parameters as a shortcut: this approach is commonly called empirical Bayes (EB). Although not rigorous from a Bayesian standpoint, the traditional folklore of EB analysis is that it provides approximations to genuine Bayesian inference, while enjoying some frequentist asymptotic guarantees. We give a new illustration of EB posterior consistency in a semiparametric estimation problem, involving the analysis of extreme multivariate events. We then drift to parametric models and focus on merging in total variation between EB and Bayesian posterior/predictive distributions, almost surely as the sample size increases. We provide new results refining those in Petrone et al. (2014) and illustrate their applications in the context of variable selection.

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