  We consider Bayesian inference of banded covariance matrices and propose a
post-processed posterior. The post-processing of the posterior consists of two
steps. In the first step, posterior samples are obtained from the conjugate
inverse-Wishart posterior which does not satisfy any structural restrictions.
In the second step, the posterior samples are transformed to satisfy the
structural restriction through a post-processing function. The conceptually
straightforward procedure of the post-processed posterior makes its computation
efficient and can render interval estimators of functionals of covariance
matrices. We show that it has nearly optimal minimax rates for banded
covariances among all possible pairs of priors and post-processing functions.
Furthermore, we prove that the expected coverage probability of the
$(1-\alpha)100\%$ highest posterior density region of the post-processed
posterior is asymptotically $1-\alpha$ with respect to a conventional posterior
distribution. It implies that the highest posterior density region of the
post-processed posterior is, on average, a credible set of a conventional
posterior. The advantages of the post-processed posterior are demonstrated by a
simulation study and a real data analysis.
