The Inverse Gamma-Gamma Prior for Optimal Posterior Contraction and Multiple Hypothesis Testing

Abstract: 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.