Generalized hyper Markov laws for directed acyclic graphs

In this paper we construct a family of DAG Wishart distributions that form a rich conjugate family of priors with multiple shape parameters for Gaussian DAG models, and proceed to undertake a theoretical analysis of this class with the goal of posterior inference. We first prove that our family of DAG Wishart distributions satisfies the strong directed hyper Markov property. Operating on the Cholesky space we derive closed form expressions for normalizing constants, posterior moments, Laplace transforms and posterior modes, and demonstrate the use of the DAG Wishart class in posterior analysis. We then consider submanifolds of the cone of positive definite matrices that correspond to covariance and concentration matrices of Gaussian DAG models. In general these spaces are curved manifolds and thus the DAG Wisharts have no density w.r.t Lebesgue measure. Hence tools for posterior inference on these spaces are not immediately available. We tackle the problem in three parts, with each part building on the previous one, until a complete solution is available for ALL DAGs.