  We study the well-known problem of estimating a sparse $n$-dimensional
unknown mean vector $\theta = (\theta_1, ..., \theta_n)$ with entries corrupted
by Gaussian white noise. In the Bayesian framework, continuous shrinkage priors
which can be expressed as scale-mixture normal densities are popular for
obtaining sparse estimates of $\theta$. In this article, we introduce a new
fully Bayesian scale-mixture prior known as the inverse gamma-gamma (IGG)
prior. We prove that the posterior distribution contracts around the true
$\theta$ at (near) minimax rate under very mild conditions. In the process, we
prove that the sufficient conditions for minimax posterior contraction given by
Van der Pas et al. (2016) are not necessary for optimal posterior contraction.
We further show that the IGG posterior density concentrates at a rate faster
than those of the horseshoe or the horseshoe+ in the Kullback-Leibler (K-L)
sense. To classify true signals ($\theta_i \neq 0$), we also propose a
hypothesis test based on thresholding the posterior mean. Taking the loss
function to be the expected number of misclassified tests, we show that our
test procedure asymptotically attains the optimal Bayes risk exactly. We
illustrate through simulations and data analysis that the IGG has excellent
finite sample performance for both estimation and classification.
