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

Abstract

In the general signal$+$noise (allowing non-normal, non-independent observations) 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 exchangeable exponential moment condition on the noise, we establish the local (oracle) optimality of the proposed confidence ball. Without EBR, we propose another confidence ball of full coverage, but its radius contains an additional $\sigma n^{1/4}$-term. In passing, we also get the local optimal results for estimation, posterior contraction problems, and the problem of weak recovery of sparsity structure. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.