Bayesian Covariate-Dependent Gaussian Graphical Models with Varying Structure

Abstract

We introduce Bayesian Gaussian graphical models with covariates (GGMx), a class of multivariate Gaussian distributions with covariate-dependent sparse precision matrix. We propose a general construction of a functional mapping from the covariate space to the cone of sparse positive definite matrices, that encompasses many existing graphical models for heterogeneous settings. Our methodology is based on a novel mixture prior for precision matrices with a non-local component that admits attractive theoretical and empirical prop- erties. The flexible formulation of GGMx allows both the strength and the sparsity pattern of the precision matrix (hence the graph structure) change with the covariates. Posterior in- ference is carried out with a carefully designed Markov chain Monte Carlo algorithm which ensures the positive definiteness of sparse precision matrices at any given covariatesâ€TM values. Extensive simulations and a case study in cancer genomics demonstrate the utility of the proposed model