  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.
