# Empirical priors and coverage of posterior credible sets in a sparse normal mean model

## TL;DR

This paper examines the frequentist validity of Bayesian credible sets in sparse normal mean models, demonstrating that empirical priors can produce credible intervals with accurate coverage under mild conditions.

## Contribution

It introduces empirical priors that ensure credible sets attain nominal coverage, even with weaker conditions than traditional selection consistency and Bernstein--von Mises theorems.

## Key findings

- Credible intervals achieve nominal frequentist coverage.
- Empirical Bayes method outperforms other fully Bayes methods in coverage.
- Conditions for coverage are weaker than those for selection consistency.

## Abstract

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.