Empirical Bayes posterior concentration in sparse high-dimensional linear models

We propose a new empirical Bayes approach for inference in the $p \gg n$ normal linear model. The novelty is the use of data in the prior in two ways, for centering and regularization. Under suitable sparsity assumptions, we establish a variety of concentration rate results for the empirical Bayes posterior distribution, relevant for both estimation and model selection. Computation is straightforward and fast, and simulation results demonstrate the strong finite-sample performance of the empirical Bayes model selection procedure.