Conditions for Posterior Contraction in the Sparse Normal Means Problem

TL;DR

This paper establishes general conditions on continuous shrinkage priors, especially scale mixtures of normals, to ensure Bayesian posterior contraction at the minimax rate in the sparse normal means problem, extending previous results.

Contribution

It provides new, verifiable conditions on the prior's tail behavior and mass distribution that guarantee optimal posterior contraction rates for a broad class of shrinkage priors.

Findings

Conditions verified for horseshoe, normal-exponential gamma, and other priors.

Extends known results to include horseshoe+ and spike-and-slab Lasso.

Guidelines for choosing priors in sparse estimation problems.

Abstract

The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as Laplace, but not too heavy, and a large amount of mass around zero relative to the tails, more so as the sparsity increases. These conditions give some general guidelines for choosing a shrinkage prior for estimation under a nearly black sparsity assumption. We verify these conditions for the class of priors considered by Ghosh and Chakrabarti (2015), which includes the horseshoe and the normal-exponential gamma priors, and for the horseshoe+, the inverse-Gaussian prior, the normal-gamma prior, and the spike-and-slab Lasso, and thus extend the number of shrinkage priors which are known to lead to posterior contraction at the minimax estimation rate.