Empirical Bayes large-scale multiple testing for high-dimensional binary outcome data

This paper explores the multiple testing problem for sparse high-dimensional data with binary outcomes. We utilize the empirical Bayes posterior to construct multiple testing procedures and evaluate their performance on false discovery rate (FDR) control. We first show that the $\ell$-value (a.k.a. the local FDR) procedure can be overly conservative in estimating the FDR if choosing the conjugate spike and uniform slab prior. To address this, we propose two new procedures that calibrate the posterior to achieve correct FDR control. Sharp frequentist theoretical results are established for these procedures, and numerical experiments are conducted to validate our theory in finite samples. To the best of our knowledge, we obtain the first {\it uniform} FDR control result in multiple testing for high-dimensional data with binary outcomes under the sparsity assumption.