The beta-mixture shrinkage prior for sparse covariances with near-minimax posterior convergence rate

Statistical inference for sparse covariance matrices is crucial to reveal the dependence structure of large multivariate data sets, but lacks scalable and theoretically supported Bayesian methods. In this paper, we propose a beta-mixture shrinkage prior, computationally more efficient than the spike and slab prior, for sparse covariance matrices and establish its minimax optimality in high-dimensional settings. The proposed prior consists of independent beta-mixture shrinkage and gamma priors for off-diagonal and diagonal entries, respectively. To ensure positive definiteness of the covariance matrix, we further restrict the support of the prior to the subspace of positive definite matrices. We obtain the posterior convergence rate of the induced posterior under the Frobenius norm and establish a minimax lower bound for sparse covariance matrices. The class of sparse covariance matrices for the minimax lower bound considered in this paper is controlled by the number of nonzero off-diagonal elements and has more intuitive appeal than those appeared in the literature. We show that the posterior convergence rates of the proposed methods are minimax or nearly minimax. In the simulation study, we also show that the proposed method is computationally more efficient than competitors while achieving comparable performance. Advantages of the beta-mixture shrinkage prior are demonstrated based on two real data sets.