Parameter priors for directed acyclic graphical models and the characterization of several probability distributions

Authors:
Dan Geiger;David Heckerman
Affiliations:
Microsoft Research, Redmond, WA;Microsoft Research, Redmond, WA
Venue:
UAI'99 Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence
Year:
1999

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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 × 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 W11 - W12W22-1, is independent of {W12, W22) for every block partitioning W11, W12, W12, W22 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.