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

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Abstract
We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x n, n >= 3, positive-definite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W_{11}-W_{12}W_{22}^{-1}W_{12}' is independent of {W_{12}, W_{22}} for every block partitioning W_{11}, W_{12}, W_{12}', W_{22} of W. Similar characterizations 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 regression model.

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