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Statistical inference for sparse covariance matrices is crucial to revealdependence structure of large multivariate data sets, but lacks scalable andtheoretically supported Bayesian methods. In this paper, we proposebeta-mixture shrinkage prior, computationally more efficient than the spike andslab prior, for sparse covariance matrices and establish its minimax optimalityin high-dimensional settings. The proposed prior consists of beta-mixtureshrinkage and gamma priors for off-diagonal and diagonal entries, respectively.To ensure positive definiteness of the resulting covariance matrix, we furtherrestrict the support of the prior to a subspace of positive definite matrices.We obtain the posterior convergence rate of the induced posterior under theFrobenius norm and establish a minimax lower bound for sparse covariancematrices. The class of sparse covariance matrices for the minimax lower boundconsidered in this paper is controlled by the number of nonzero off-diagonalelements and has more intuitive appeal than those appeared in the literature.The obtained posterior convergence rate coincides with the minimax lower boundunless the true covariance matrix is extremely sparse. In the simulation study,we show that the proposed method is computationally more efficient thancompetitors, while achieving comparable performance. Advantages of theshrinkage prior are demonstrated based on two real data sets.

The Beta-Mixture Shrinkage Prior for Sparse Covariances with Posterior Minimax Rates