  We present an objective Bayes method for covariance selection in Gaussian
multivariate regression models whose error term has a covariance structure
which is Markov with respect to a Directed Acyclic Graph (DAG). The scope is
covariate-adjusted sparse graphical model selection, a topic of growing
importance especially in the area of genetical genomics (eQTL analysis).
Specifically, we provide a closed-form expression for the marginal likelihood
of any DAG (with small parent sets) whose computation virtually requires no
subjective elicitation by the user and involves only conjugate matrix normal
Wishart distributions. This is made possible by a specific form of prior
assignment, whereby only one prior under the complete DAG model need be
specified, based on the notion of fractional Bayes factor. All priors under the
other DAG models are derived using prior modularity, and global parameter
independence, in the terminology of Geiger & Heckerman (2002). Since the
marginal likelihood we obtain is constant within each class of Markov
equivalent DAGs, our method naturally specializes to covariate-adjusted
decomposable graphical models.
