We revisit the problem of simultaneously testing the means of @math 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 @math is known. When @math 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 @math . We further introduce 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. Numerical experiments strongly suggest that adaptive test procedures based on REML and hierarchical Bayes considerations also have the oracle property for multiple testing. We illustrate our methodology through simulations and analysis of a prostate cancer data set.
Under certain conditions on the prior for the scale parameter @math in ), @cite_18 derived asymptotic upper bounds on Type I and Type II errors and the Bayes risk ) for both non-adaptive and data-adaptive test procedures induced by scale-mixture shrinkage priors. Under these conditions, the upper bound on the Bayes risk for scale-mixture priors is of the same order as the Bayes Oracle risk. Specifically, @cite_18 showed that the Bayes risk ) for thresholding rule ) can be bounded from above by @math for known @math and by @math for unknown @math , where @math is the constant from the fourth condition in Assumption 1 and @math are appropriate constants that depend on the prior.
@cite_18
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In this paper we study the asymptotic properties of Bayesian multiple testing procedures for a large class of Gaussian scale mixture pri- ors. We study two types of multiple testing risks: a Bayesian risk proposed in (2011) where the data are assume to come from a mixture of normal, and a frequentist risk similar to the one proposed by Arias-Castro and Chen (2017). Following the work of van der (2016), we give general conditions on the prior such that both risks can be bounded. For the Bayesian risk, the bound is almost sharp. This result show that under these conditions, the considered class of continuous prior can be competitive with the usual two-group model (e.g. spike and slab priors). We also show that if the non-zeros component of the parameter are large enough, the minimax risk can be made asymptotically null. The separation rates obtained are consistent with the one that could be guessed from the existing literature (see van der , 2017b). For both problems, we then give conditions under which an adaptive version of the result can be obtained.