Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector

TL;DR

This paper introduces a new empirical Bayes approach for estimating sparse high-dimensional normal mean vectors, achieving asymptotic minimaxity and effective support recovery, with promising simulation results.

Contribution

It proposes a novel empirical Bayes model that ensures the posterior concentrates at the minimax rate and accurately recovers the support of the sparse mean vector.

Findings

Posterior concentrates on minimax balls around the true mean.

Posterior mean is asymptotically minimax.

Support of the posterior matches the true sparsity pattern.

Abstract

For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mean vector. Simulation from our empirical Bayes posterior is straightforward, and our numerical results demonstrate the quality of our method compared to others having similar large-sample properties.