Empirical Bayes posterior concentration in sparse high-dimensional linear models

We propose an 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, our empirical Bayes posterior distribution for the regression coefficients concentrates at the frequentist minimax rate for suitable priors on the model size. Model selection consistency is established, and simulation studies show the strong finite-sample performance of our method.