On coverage and local radial rates of credible sets

Abstract

In the mildly ill-posed inverse signal-in-white-noise model, we construct confidence sets as credible balls with respect to the empirical Bayes posterior resulting from a certain two-level hierarchical prior. The quality of the posterior is characterized by the contraction rate which we allow to be local, that is, depending on the parameter. The issue of optimality of the constructed confidence sets is addressed via a trade-off between its “size” (the local radial rate) and its coverage probability. We introduce excessive bias restriction (EBR), more general than self-similarity and polished tail condition recently studied in the literature. Under EBR, we establish the confidence optimality of our credible set with some local (oracle) radial rate. We also derive the oracle estimation inequality and the oracle posterior contraction rate. The obtained local results are more powerful than global: adaptive minimax results for a number of smoothness scales follow as consequence, in particular, the ones considered by Szabó et al. [Ann. Statist. 43 (2015) 1391–1428].