Empirical Bayes posterior concentration in sparse high-dimensional linear models

TL;DR

This paper introduces an empirical Bayes method for high-dimensional linear models that leverages data in the prior, achieving fast computation and strong finite-sample performance for estimation and model selection.

Contribution

It presents a novel empirical Bayes approach that incorporates data into the prior for sparse high-dimensional linear models, with proven concentration rates.

Findings

Achieves favorable posterior concentration rates under sparsity.

Demonstrates strong finite-sample performance in simulations.

Provides a computationally straightforward and fast method.

Abstract

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.