  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.
