Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions
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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_12W_22^-1W_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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