  We consider Bayesian estimation of a $p\times p$ precision matrix, when $p$
can be much larger than the available sample size $n$. It is well known that
consistent estimation in such ultra-high dimensional situations requires
regularization such as banding, tapering or thresholding. We consider a banding
structure in the model and induce a prior distribution on a banded precision
matrix through a Gaussian graphical model, where an edge is present only when
two vertices are within a given distance. For a proper choice of the order of
graph, we obtain the convergence rate of the posterior distribution and Bayes
estimators based on the graphical model in the $L_{\infty}$-operator norm
uniformly over a class of precision matrices, even if the true precision matrix
may not have a banded structure. Along the way to the proof, we also compute
the convergence rate of the maximum likelihood estimator (MLE) under the same
set of condition, which is of independent interest. The graphical model based
MLE and Bayes estimators are automatically positive definite, which is a
desirable property not possessed by some other estimators in the literature. We
also conduct a simulation study to compare finite sample performance of the
Bayes estimators and the MLE based on the graphical model with that obtained by
using a Cholesky decomposition of the precision matrix. Finally, we discuss a
practical method of choosing the order of the graphical model using the
marginal likelihood function.
