Posterior convergence rates for estimating large precision matrices using Graphical models

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. We show that under a very mild growth condition and a proper choice of the order of graph, the posterior distribution based on the graphical model is consistent 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 establish that the maximum likelihood estimator (MLE) is also consistent under the same set of condition, which is of independent interest. We also conduct a simulation study to compare finite sample performance of the Bayes estimator and the MLE based on the graphical model with that obtained by using a banding operation on the sample covariance matrix. We observe that the graphical model based estimators perform significantly better, especially if the banded sample covariance matrix is not positive definite. In contrast, the graphical model based estimators are always positive definite. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.