Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences

We consider full Bayesian inference in the multivariate normal mean model in the situation that the mean vector is sparse. The prior distribution on the vector of means is constructed hierarchically by first choosing a collection of nonzero means and next a prior on the nonzero values. We consider the posterior distribution in the frequentist set-up that the observations are generated according to a fixed mean vector, and are interested in the posterior distribution of the number of nonzero components and the contraction of the posterior distribution to the true mean vector. We find various combinations of priors on the number of nonzero coefficients and on these coefficients that give desirable performance. We also find priors that give suboptimal convergence, for instance, Gaussian priors on the nonzero coefficients. We illustrate the results by simulations.