High dimensional Bayesian inference for Gaussian directed acyclic graph models

In this paper, we consider Gaussian models Markov with respect to an arbitrary DAG. We first construct a family of conjugate priors for the Cholesky parametrization of the covariance matrix of such models. This family has as many shape parameters as the DAG has vertices, and naturally extends the work of Geiger and Heckerman [8]. From these distributions, we derive prior distributions for the covariance and precision parameters of the Gaussian DAG Markov models. Our works thus extends the work of Dawid and Lauritzen [5] and Letac and Massam [16] for Gaussian models Markov with respect to a decomposable graph to arbitrary DAGs. For this reason, we call our distributions DAG-Wishart distributions. An advantage of these distributions is that they possess strong hyper Markov properties and thus allow for explicit estimation of the covariance and precision parameters, regardless of the dimension of the problem. They also allow us to develop methodology for model selection and covariance estimation in the space of DAG-Markov models. We demonstrate via several numerical examples that the proposed method scales well to high-dimensions.