Large-scale multiple hypothesis testing with the normal-beta prime prior

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 studying a scale-mixture prior known as the normal-beta prime (NBP) prior. To detect signals, we 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 $p\propto n^{-ϵ},ϵ\in (0,1)$. 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.

KEYWORDS:

• Bayes oracle
• empirical Bayes
• multiple testing
• shrinkage prior
• sparsity

Acknowledgements

The authors would like to thank Dr. Anirban Bhattacharya and Dr. Xueying Tang for sharing their codes, which were modified to generate Figures 1–5. We are grateful to two anonymous reviewers, the Associate Editor, and the Editors whose thoughtful comments and suggestions helped to greatly improve this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 For the method $\tau \sim \mathcal{U}(1/n,1)$, we slightly modify the code in the HS.normal.means function in the horseshoe R package.