Large-Scale Multiple Hypothesis Testing with the Normal-Beta Prime Prior

TL;DR

This paper introduces a Bayesian multiple testing method using the normal-beta prime prior, achieving asymptotic optimality in sparse normal mean problems and extending to empirical Bayes and hierarchical approaches.

Contribution

It develops a new Bayesian testing procedure with the NBP prior, deriving concentration properties and demonstrating asymptotic optimality under sparsity.

Findings

The proposed test asymptotically attains the Bayes risk when the signal proportion is known.

The empirical Bayes variant also achieves the Bayes Oracle risk across various sparsity levels.

Hierarchical Bayes and REML methods for hyperparameter estimation are examined.

Abstract

We revisit the problem of simultaneously testing the means of $n$ independent normal observations under sparsity. We take a Bayesian approach to this problem by introducing a scale-mixture prior known as the normal-beta prime (NBP) prior. We first derive new concentration properties when the beta prime density is employed for a scale parameter in Bayesian hierarchical models. To detect signals in our data, we then propose a hypothesis test based on thresholding the posterior shrinkage weight under the NBP prior. Taking the loss function to be the expected number of misclassified tests, we show that our test procedure asymptotically attains the optimal Bayes risk when the signal proportion $p$ is known. When $p$ is unknown, we introduce an empirical Bayes variant of our test which also asymptotically attains the Bayes Oracle risk in the entire range of sparsity parameters $p \propto n^{-\epsilon}, \epsilon \in (0, 1)$. Finally, we also consider restricted marginal maximum likelihood (REML) and hierarchical Bayes approaches for estimating a key hyperparameter in the NBP prior and examine multiple testing under these frameworks.