Bayesian Inference for Gaussian Graphical Models Beyond Decomposable Graphs

Abstract

Summary Bayesian inference for graphical models has received much attention in the literature in recent years. It is well known that, when the graph G is decomposable, Bayesian inference is significantly more tractable than in the general non-decomposable setting. Penalized likelihood inference in contrast has made tremendous gains in the past few years in terms of scalability and tractability. Bayesian inference, however, has not had the same level of success, though a scalable Bayesian approach has its strengths, especially in terms of quantifying uncertainty. To address this gap, we propose a scalable and flexible novel Bayesian approach for estimation and model selection in Gaussian undirected graphical models. We first develop a class of generalized G-Wishart distributions with multiple shape parameters for an arbitrary underlying graph. This class contains the G-Wishart distribution as a special case. We then introduce the class of generalized Bartlett graphs and derive an efficient Gibbs sampling algorithm to obtain posterior draws from generalized G-Wishart distributions corresponding to a generalized Bartlett graph. The class of generalized Bartlett graphs contains the class of decomposable graphs as a special case but is substantially larger than the class of decomposable graphs. We proceed to derive theoretical properties of the proposed Gibbs sampler. We then demonstrate that the proposed Gibbs sampler is scalable to significantly higher dimensional problems compared with using an accept–reject or a Metropolis–Hasting algorithm. Finally, we show the efficacy of the proposed approach on simulated and real data. In particular, we demonstrate that our generalized Bartlett methodology can be used for efficient model selection by reducing the graph search space by using penalized likelihood and pseudolikelihood methods.