  Bayesian methods provide a natural means for uncertainty quantification, that
is, credible sets can be easily obtained from the posterior distribution. But
is this uncertainty quantification valid in the sense that the posterior
credible sets attain the nominal frequentist coverage probability? This paper
investigates the frequentist validity of posterior uncertainty quantification
based on a class of empirical priors in the sparse normal mean model. In
particular, we show that our marginal posterior credible intervals achieve the
nominal frequentist coverage probability under conditions slightly weaker than
needed for selection consistency and a Bernstein--von Mises theorem for the
full posterior, and numerical investigations suggest that our empirical Bayes
method has superior frequentist coverage probability properties compared to
other fully Bayes methods.
