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. Assuming the regression coefficients are sparse, in the sense that no more than $n$ of them are non-zero, we provide a variety of concentration rate results for our empirical Bayes posterior distribution, relevant for both estimation and model selection. Simulation results demonstrate the strong finite-sample performance of our empirical Bayes model selection procedure.