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We consider the joint sparse estimation of regression coefficients and thecovariance matrix for covariates in a high-dimensional regression model, wherethe predictors are both relevant to a response variable of interest andfunctionally related to one another via a Gaussian directed acyclic graph (DAG)model. Gaussian DAG models introduce sparsity in the Cholesky factor of theinverse covariance matrix, and the sparsity pattern in turn corresponds tospecific conditional independence assumptions on the underlying predictors. Avariety of methods have been developed in recent years for Bayesian inferencein identifying such network-structured predictors in regression setting, yetcrucial sparsity selection properties for these models have not been thoroughlyinvestigated. In this paper, we consider a hierarchical model with spike andslab priors on the regression coefficients and a flexible and general class ofDAG-Wishart distributions with multiple shape parameters on the Choleskyfactors of the inverse covariance matrix. Under mild regularity assumptions, weestablish the joint selection consistency for both the variable and theunderlying DAG of the covariates when the dimension of predictors is allowed togrow much larger than the sample size. We demonstrate that our methodoutperforms existing methods in selecting network-structured predictors inseveral simulation settings.

Joint Bayesian Variable and DAG Selection Consistency for High-dimensional Regression Models with Network-structured Covariates