Bayesian estimation of a sparse precision matrix

TL;DR

This paper introduces a Bayesian method for estimating sparse precision matrices in high-dimensional Gaussian models, using mixture priors and Laplace approximation for efficient graph structure learning.

Contribution

It develops a Bayesian framework with mixture priors for sparse precision matrix estimation and proposes a fast Laplace approximation method for posterior computation.

Findings

Posterior convergence rate for the precision matrix established.

Laplace approximation effectively estimates posterior graph probabilities.

Method performs well in high-dimensional settings.

Abstract

We consider the problem of estimating a sparse precision matrix of a multivariate Gaussian distribution, including the case where the dimension $p$ is large. Gaussian graphical models provide an important tool in describing conditional independence through presence or absence of the edges in the underlying graph. A popular non-Bayesian method of estimating a graphical structure is given by the graphical lasso. In this paper, we consider a Bayesian approach to the problem. We use priors which put a mixture of a point mass at zero and certain absolutely continuous distribution on off-diagonal elements of the precision matrix. Hence the resulting posterior distribution can be used for graphical structure learning. The posterior convergence rate of the precision matrix is obtained. The posterior distribution on the model space is extremely cumbersome to compute. We propose a fast computational method for approximating the posterior probabilities of various graphs using the Laplace approximation approach by expanding the posterior density around the posterior mode, which is the graphical lasso by our choice of the prior distribution. We also provide estimates of the accuracy in the approximation.