Multiple hypothesis testing is a topic of growing importance in statistics, particularly for the analysis of high-dimensional data. For example, in microarray experiments, thousands of hypothesis tests are performed simultaneously in order to identify significant genes that are associated with some biological trait of interest.Other applications include identifying hydrocarbon from seismic data, detecting moving objects in radar detection, and finding tumors in medical imaging. This testing problem typically falls within the framework of testing for the means of independent normal observations to determine which means differ significantly from zero.

We consider the multiple testing problem in the Bayesian domain by placing global-local shrinkage priors on the normal means and determining an appropriate thresholding rule for rejecting (or failing to reject) the null hypothesis. These  shrinkage priors contain a global parameter (tau) common to all observations which shrinks all posterior means to zero, and a local parameter specific to each observation — typically from a heavy-tailed prior distribution — that either shrinks the posterior mean further to zero or that cancels out the shrinkage by the global parameter, depending on how far away the observation is from zero.  In the literature, tau has traditionally been treated as a tuning parameter or as an empirical Bayes plug-in estimator. In this talk, we instead consider a fully hierarchical Bayesian model by placing a prior on tau, which allows the global parameter to “learn” from the data. We provide sufficient conditions for the prior on tau so that our testing procedure enjoys asymptotic optimality properties in terms of Bayes risk under assumptions of sparsity.

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