Local posterior concentration rate for multilevel sparse sequences

We consider empirical Bayesian inference in the many normal means model in the situation when the high-dimensional mean vector is multilevel sparse, that is, most of the entries of the parameter vector are some fixed values. For instance, the traditional sparse signal is a particular case (with one level) of multilevel sparse sequences. We apply an empirical Bayesian approach, namely we put an appropriate prior modeling the multilevel sparsity and make data-dependent choices of certain parameters of the prior. We establish local (i.e., with rate depending on the “true” parameter) posterior contraction and estimation results. Global adaptive minimax results (for the estimation and posterior contraction problems) over sparsity classes follow from our local results if the sparsity level is of polynomial order. The results are illustrated by simulations.