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