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Parameter Priors for Directed Acyclic Gr...
Geiger, Dan...

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Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions by Geiger, Dan ( Author )
Lemonjar Open Access
Australian National University
01-08-2023
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_-W_W_{22}^{-1}W_' is independent of {W_, W_{22}} for every block partitioning W_, W_, W_', 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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http://arxiv.org/abs/1301.6697
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