  Covariance estimation and selection for multivariate datasets in a
high-dimensional regime is a fundamental problem in modern statistics. Gaussian
graphical models are a popular class of models used for this purpose. Current
Bayesian methods for inverse covariance matrix estimation under Gaussian
graphical models require the underlying graph and hence the ordering of
variables to be known. However, in practice, such information on the true
underlying model is often unavailable. We therefore propose a novel
permutation-based Bayesian approach to tackle the unknown variable ordering
issue. In particular, we utilize multiple maximum a posteriori estimates under
the DAG-Wishart prior for each permutation, and subsequently construct the
final estimate of the inverse covariance matrix. The proposed estimator has
smaller variability and yields order-invariant property. We establish posterior
convergence rates under mild assumptions and illustrate that our method
outperforms existing approaches in estimating the inverse covariance matrices
via simulation studies.
