Empirical priors and coverage of posterior credible sets in a sparse normal mean model
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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 validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. We prove that there are scenarios in which the empirical Bayes method provides valid uncertainty quantification while other methods may not, and finite-sample simulations confirm the asymptotic findings.
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