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In the sparse sequence model, we consider a popular Bayesian multiple testingprocedure and investigate for the first time its behaviour from the frequentistpoint of view. Given a spike-and-slab prior on the high-dimensional sparseunknown parameter, one can easily compute posterior probabilities of comingfrom the spike, which correspond to the well known local-fdr values, alsocalled $\ell$-values. The spike-and-slab weight parameter is calibrated in anempirical Bayes fashion, using marginal maximum likelihood. The multipletesting procedure under study, called here the cumulative $\ell$-valueprocedure, ranks coordinates according to their empirical $\ell$-values andthresholds so that the cumulative ranked sum does not exceed a user-specifiedlevel $t$.  We validate the use of this method from the multiple testing perspective: foralternatives of appropriately large signal strength, the false discovery rate(FDR) of the procedure is shown to converge to the target level $t$, while itsfalse negative rate (FNR) goes to $0$. We complement this study by providingconvergence rates for the method. Additionally, we prove that the $q$-valuemultiple testing procedure shares similar convergence rates in this model.

Empirical Bayes cumulative $\ell$-value multiple testing procedure for sparse sequences