Bayesian inference for high-dimensional decomposable graphs

Abstract

In this paper, we consider high-dimensional Gaussian graphical models where the true underlying graph is decomposable. A hierarchical G-Wishart prior is proposed to conduct a Bayesian inference for the precision matrix and its graph structure. Although the posterior asymptotics using the G-Wishart prior has received increasing attention in recent years, most of the results assume moderate high-dimensional settings, where the number of variables p is smaller than the sample size n. However, this assumption might not hold in many real applications such as genomics, speech recognition and climatology. Motivated by this gap, we investigate asymptotic properties of posteriors under the high-dimensional setting where p can be much larger than n. The pairwise Bayes factor consistency, posterior ratio consistency and graph selection consistency are obtained in this high-dimensional setting. Furthermore, the posterior convergence rate for precision matrices under the matrix ${\ell }_1$-norm is derived, which is faster than posterior convergence rates obtained in existing literature. A simulation study confirms that the proposed Bayesian procedure outperforms competitors.