Posterior convergence rates for estimating large precision matrices using graphical models

TL;DR

This paper develops Bayesian methods for estimating large precision matrices with a banded structure, providing convergence rates and practical algorithms for high-dimensional graphical models.

Contribution

It introduces a Bayesian approach with a banded graphical model prior, deriving convergence rates for the posterior and estimators in ultra-high dimensional settings.

Findings

Posterior convergence rate established for high-dimensional precision matrices.

Graphical model-based estimators are positive definite and perform well in simulations.

A practical method for selecting the model order using marginal likelihood is proposed.

Abstract

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.