Spike and slab empirical Bayes sparse credible sets

TL;DR

This paper investigates the coverage properties of adaptive Bayesian credible sets in sparse normal means models using spike and slab priors, establishing contraction rates and optimal credible sets under certain conditions.

Contribution

It introduces a method for calibrating the sparsity hyperparameter via empirical Bayes and constructs credible sets with guaranteed coverage and optimal size.

Findings

Derived adaptive posterior contraction rates for $d_q$--distances.

Constructed credible sets with coverage at the prescribed confidence level.

Proved the necessity of certain conditions for minimax optimality.

Abstract

In the sparse normal means model, coverage of adaptive Bayesian posterior credible sets associated to spike and slab prior distributions is considered. The key sparsity hyperparameter is calibrated via marginal maximum likelihood empirical Bayes. First, adaptive posterior contraction rates are derived with respect to $d_q$--type--distances for $q\leq 2$. Next, under a type of so-called excessive-bias conditions, credible sets are constructed that have coverage of the true parameter at prescribed $1-\alpha$ confidence level and at the same time are of optimal diameter. We also prove that the previous conditions cannot be significantly weakened from the minimax perspective.