  Statistical inference for sparse covariance matrices is crucial to reveal
dependence structure of large multivariate data sets, but lacks scalable and
theoretically supported Bayesian methods. In this paper, we propose
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 beta-mixture
shrinkage and gamma priors for off-diagonal and diagonal entries, respectively.
To ensure positive definiteness of the resulting covariance matrix, we further
restrict the support of the prior to a 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.
The obtained posterior convergence rate coincides with the minimax lower bound
unless the true covariance matrix is extremely sparse. In the simulation study,
we show that the proposed method is computationally more efficient than
competitors, while achieving comparable performance. Advantages of the
shrinkage prior are demonstrated based on two real data sets.
