Bayesian Covariate-Dependent Gaussian Graphical Models with Varying Structure

We introduce Bayesian Gaussian graphical models with covariates (GGMx), a class ofmultivariate Gaussian distributions with covariate-dependent sparse precision matrix. Wepropose a general construction of a functional mapping from the covariate space to the coneof sparse positive definite matrices, which encompasses many existing graphical models forheterogeneous settings. Our methodology is based on a novel mixture prior for precisionmatrices 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 patternof the precision matrix (hence the graph structure) change with the covariates. Posteriorinference is carried out with a carefully designed Markov chain Monte Carlo algorithm,which ensures the positive definiteness of sparse precision matrices at any given covariates’values. Extensive simulations and a case study in cancer genomics demonstrate the utilityof the proposed model