  We consider the joint sparse estimation of regression coefficients and the
covariance matrix for covariates in a high-dimensional regression model, where
the predictors are both relevant to a response variable of interest and
functionally related to one another via a Gaussian directed acyclic graph (DAG)
model. Gaussian DAG models introduce sparsity in the Cholesky factor of the
inverse covariance matrix, and the sparsity pattern in turn corresponds to
specific conditional independence assumptions on the underlying predictors. A
variety of methods have been developed in recent years for Bayesian inference
in identifying such network-structured predictors in regression setting, yet
crucial sparsity selection properties for these models have not been thoroughly
investigated. In this paper, we consider a hierarchical model with spike and
slab priors on the regression coefficients and a flexible and general class of
DAG-Wishart distributions with multiple shape parameters on the Cholesky
factors of the inverse covariance matrix. Under mild regularity assumptions, we
establish the joint selection consistency for both the variable and the
underlying DAG of the covariates when the dimension of predictors is allowed to
grow much larger than the sample size. We demonstrate that our method
outperforms existing methods in selecting network-structured predictors in
several simulation settings.
