Needles and straw in a haystack: robust confidence for possibly sparse sequences

TL;DR

This paper develops a robust Bayesian method for uncertainty quantification in sparse signal estimation, introducing a new bias restriction condition that ensures local optimality and adaptivity across sparsity classes.

Contribution

It introduces the excessive bias restriction (EBR) condition and constructs an empirical Bayes posterior for reliable uncertainty quantification in sparse models.

Findings

Confidence balls are locally optimal under EBR.

The method achieves adaptive minimax rates.

Results include local optimality for estimation and posterior contraction.

Abstract

In the general signal+noise model we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel excessive bias restriction (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild conditions on the noise, we establish the local (oracle) optimality of the proposed confidence ball. In passing, we also get the local optimal (oracle) results for estimation and posterior contraction problems. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.