Large-Scale Multiple Hypothesis Testing with the Normal-Beta Prime Prior
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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 ∝ n^-ϵ, ϵ∈ (0, 1). We further consider a restricted marginal maximum likelihood (REML) approach for estimating a key hyperparameter in the NBP prior and examine multiple testing under this framework. Numerical experiments strongly suggest that adaptive test procedures based on REML also have the oracle property for multiple testing. We illustrate our methodology through simulations and analysis of a prostate cancer data set.
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