  Shrinkage prior are becoming more and more popular in Bayesian modeling for
high dimensional sparse problems due to its computational efficiency. Recent
works show that a polynomially decaying prior leads to satisfactory posterior
asymptotics under regression models. In the literature, statisticians have
investigated how the global shrinkage parameter, i.e., the scale parameter, in
a heavy tail prior affects the posterior contraction. In this work, we explore
how the shape of the prior, or more specifically, the polynomial order of the
prior tail affects the posterior. We discover that, under the sparse normal
means models, the polynomial order does affect the multiplicative constant of
the posterior contraction rate. More importantly, if the polynomial order is
sufficiently close to 1, it will induce the optimal Bayesian posterior
convergence, in the sense that the Bayesian contraction rate is sharply
minimax, i.e., not only the order, but also the multiplicative constant of the
posterior contraction rate are optimal. The above Bayesian sharp minimaxity
holds when the global shrinkage parameter follows a deterministic choice which
depends on the unknown sparsity $s$. Therefore, a Beta-prior modeling is
further proposed, such that our sharply minimax Bayesian procedure is adaptive
to unknown $s$. Our theoretical discoveries are justified by simulation
studies.
