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Gaussian graphical models typically assume a homogeneous structure across allsubjects, which is often restrictive in applications. In this article, wepropose a weighted pseudo-likelihood approach for graphical modeling whichallows different subjects to have different graphical structures depending onextraneous covariates. The pseudo-likelihood approach replaces the jointdistribution by a product of the conditional distributions of each variable. Wecast the conditional distribution as a heteroscedastic regression problem, withcovariate-dependent variance terms, to enable information borrowing directlyfrom the data instead of a hierarchical framework. This allows independentgraphical modeling for each subject, while retaining the benefits of ahierarchical Bayes model and being computationally tractable. An efficientembarrassingly parallel variational algorithm is developed to approximate theposterior and obtain estimates of the graphs. Using a fractional variationalframework, we derive asymptotic risk bounds for the estimate in terms of anovel variant of the $\alpha$-R\'{e}nyi divergence. We theoreticallydemonstrate the advantages of information borrowing across covariates overindependent modeling. We show the practical advantages of the approach throughsimulation studies and illustrate the dependence structure in proteinexpression levels on breast cancer patients using CNV information ascovariates.

An Approximate Bayesian Approach to Covariate-dependent Graphical Modeling