Posterior Contraction rate and Asymptotic Bayes Optimality for one-group shrinkage priors in sparse normal means problem

TL;DR

This paper investigates the asymptotic properties of one-group global-local shrinkage priors, including the horseshoe prior, in high-dimensional sparse normal means models, establishing near-minimax posterior contraction rates and asymptotic Bayes optimality.

Contribution

It extends previous results by analyzing a broad class of priors, including hierarchical Bayes and empirical Bayes approaches, proving near-minimax contraction and Bayes optimality in sparse settings.

Findings

Posterior contracts at near-minimax rate under empirical Bayes.

Hierarchical Bayes estimates achieve asymptotic minimax risk.

Conditions are identified for global parameter priors to attain asymptotic Bayes optimality.

Abstract

We consider a high-dimensional sparse normal means model where the goal is to estimate the mean vector assuming the proportion of non-zero means is unknown. We model the mean vector by a one-group global-local shrinkage prior belonging to a broad class of such priors that includes the horseshoe prior. We address some questions related to asymptotic properties of the resulting posterior distribution of the mean vector for the said class priors. We consider two ways to model the global parameter in this paper. Firstly by considering this as an unknown fixed parameter and then by an empirical Bayes estimate of it. In the second approach, we do a hierarchical Bayes treatment by assigning a suitable non-degenerate prior distribution to it. We first show that for the class of priors under study, the posterior distribution of the mean vector contracts around the true parameter at a near minimax rate when the empirical Bayes approach is used. Next, we prove that in the hierarchical Bayes approach, the corresponding Bayes estimate attains the minimax risk asymptotically under the squared error loss function. We also show that the posterior contracts around the true parameter at a near minimax rate. These results generalize those of van der Pas et al. (2014) \cite{van2014horseshoe}, (2017) \cite{van2017adaptive}, proved for the horseshoe prior. We have also studied in this work the asymptotic Bayes optimality of global-local shrinkage priors where the number of non-null hypotheses is unknown. Here our target is to propose some conditions on the prior density of the global parameter such that the Bayes risk induced by the decision rule attains Optimal Bayes risk, up to some multiplicative constant. Using our proposed condition, under the asymptotic framework of Bogdan et al. (2011) \cite{bogdan2011asymptotic}, we are able to provide an affirmative answer to satisfy our hunch.