On Bayesian based adaptive confidence sets for linear functionals

TL;DR

This paper investigates Bayesian confidence sets for linear functionals in inverse Gaussian white noise models, showing that under self-similarity assumptions, these sets achieve adaptive size and optimal coverage, with practical construction of credible bands.

Contribution

It introduces a data-driven empirical Bayes method for hyper-parameter selection and demonstrates its effectiveness under self-similarity assumptions for adaptive confidence sets.

Findings

Credible sets have sub-optimal behavior generally.

Under self-similarity, credible sets are rate-adaptive and optimally cover.

Constructs adaptive $L_{}$-credible bands with optimal coverage.

Abstract

We consider the problem of constructing Bayesian based confidence sets for linear functionals in the inverse Gaussian white noise model. We work with a scale of Gaussian priors indexed by a regularity hyper-parameter and apply the data-driven (slightly modified) marginal likelihood empirical Bayes method for the choice of this hyper-parameter. We show by theory and simulations that the credible sets constructed by this method have sub-optimal behaviour in general. However, by assuming "self-similarity" the credible sets have rate-adaptive size and optimal coverage. As an application of these results we construct $L_{\infty}$-credible bands for the true functional parameter with adaptive size and optimal coverage under self-similarity constraint.