Prior Distributions for Objective Bayesian Analysis

We provide a review of prior distributions for objective Bayesian analysis. We start by examining some foundational issues and then organize our exposition into priors for: i) estimation or prediction; ii) model selection; iii) high-dimensional models. With regard to i), we present some basic notions, and then move to more recent contributions on discrete parameter space, hierarchical models, nonparametric models, and penalizing complexity priors. Point ii) is the focus of this paper: it discusses principles for objective Bayesian model comparison, and singles out some major concepts for building priors, which are subsequently illustrated in some detail for the classic problem of variable selection in normal linear models. We also present some recent contributions in the area of objective priors on model space. With regard to point iii) we only provide a short summary of some default priors for high-dimensional models, a rapidly growing area of research.