Optimal Bayesian posterior concentration rates with empirical priors

In high-dimensional Bayesian applications, choosing a prior such that the corresponding posterior distribution has optimal asymptotic concentration properties can be restrictive in the sense that the priors supported by theory may not be computationally convenient, and vice versa. This paper develops a general strategy for constructing empirical or data-dependent priors whose corresponding posterior distributions have optimal concentration rate properties. The basic idea is to center the prior in a specific way on a good estimator. This makes the asymptotic properties of the posterior less sensitive to the shape of the prior which, in turn, allows users to work with priors of convenient forms while maintaining the optimal posterior concentration rates. General results on both adaptive and non-adaptive rates based on empirical priors are presented, along with illustrations in density estimation, nonparametric regression, and high-dimensional structured normal models.