Empirical priors for targeted posterior concentration rates

In high-dimensional problems, choosing a prior distribution such that the corresponding posterior has desirable properties can be challenging. This paper develops a general strategy for constructing empirical or data-dependent priors whose corresponding posterior distributions achieve targeted, often optimal, concentration rates. The idea is to place a prior which has sufficient mass on parameter values for which the likelihood is suitably large. 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 desired 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.