Asymptotic frequentist coverage properties of Bayesian credible sets for sieve priors

TL;DR

This paper studies the frequentist coverage of Bayesian credible sets in nonparametric models, introducing a 'general polished tail' condition to ensure honest, adaptive confidence sets with near minimax size.

Contribution

It proposes a new assumption enabling honest adaptive confidence sets for sieve priors and applies it to various models, demonstrating near minimax optimality.

Findings

Hierarchical and empirical Bayes methods produce honest confidence sets under the new condition.

The size of the confidence sets is near minimax adaptive in several models.

The approach applies to nonparametric regression, density estimation, and classification.

Abstract

We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called "general polished tail" condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abstract results to various examples, including the nonparametric regression model, density estimation using exponential families of priors, density estimation using histogram priors and nonparametric classification model, for which we show that their size is near minimax adaptive with respect to the considered specific semi-metrics.