  There has been an intense development of Bayes graphical model estimation
approaches over the past decade - however, most of the existing methods are
restricted to moderate dimensions. We propose a novel approach suitable for
high dimensional settings, by decoupling model fitting and covariance
selection. First, a full model based on a complete graph is fit under novel
class of continuous shrinkage priors on the precision matrix elements, which
induces shrinkage under an equivalence with Cholesky-based regularization while
enabling conjugate updates of entire precision matrices. Subsequently, we
propose a post-fitting graphical model estimation step which proceeds using
penalized joint credible regions to perform neighborhood selection sequentially
for each node. The posterior computation proceeds using straightforward fully
Gibbs sampling, and the approach is scalable to high dimensions. The proposed
approach is shown to be asymptotically consistent in estimating the graph
structure for fixed $p$ when the truth is a Gaussian graphical model.
Simulations show that our approach compares favorably with Bayesian competitors
both in terms of graphical model estimation and computational efficiency. We
apply our methods to high dimensional gene expression and microRNA datasets in
cancer genomics.
