Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions

We show that the only parameter prior for complete Gaussian DAG models that satis­ fies global parameter independence, complete model equivalence, and some weak regular­ ity assumptions, is the normal-Wishart dis­ tribution. Our analysis is based on the fol­ lowing new characterization of the Wishart distribution: let W be an n x n, n 2: 3, positive-definite symmetric matrix of ran­ dom variables and J(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W11 W1 2 W22 1 W{2 is independent of {W12 , W2 2 } for every block partitioning Wu, W12, W{2, W2 2 of W. Similar character­ izations of the normal and normal-Wishart distributions are provided as well. We also show how to construct a prior for every DAG model over X from the prior of a single re­ gression model.