Empirical Bayes methods in high dimensions: a survey and ongoing debates

Bayesian inference has as its starting point the specification of a prior distribution on the (possibly infinite-dimensional) parameters of the adopted statistical model. In some cases, a specification genuinely based on information available a priori and formalizing one’s level of uncertainty is difficult. This is especially true for the high-dimensional models in use for complex, recent applications of statistics and machine learning. In such circumstances, a popular practice, known as empirical Bayes, is to fix the value of the most relevant prior hyperparameters through the data. Notable examples are hyperparameters controlling sparsity in linear regression or sequence models; complexity in model selection or neural networks architecture; smoothness in nonparametric regression or density estimation. In spite of their popularity, empirical Bayesian methods still raise concerns of degeneracies and poor uncertainty quantification on the part of some scholars and practitioners, especially when set against fully Bayesian methods, whereby a hyperprior is specified on hyperparameters. The aim of this paper is to bring clarity by providing a critical review of recent advances in empirical Bayes methods for high-dimensional analysis. We offer an overview of their theoretical properties using the notion of oracle priors, illustrating with examples different facets of posterior adaptation. Finally, we discuss open issues, actively researched topics and future prospects.