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October 2002 Parameter priors for directed acyclic graphical models and the characterization of several probability distributions
Dan Geiger, David Heckerman
Ann. Statist. 30(5): 1412-1440 (October 2002). DOI: 10.1214/aos/1035844981


We develop simple methods for constructing parameter priors for model choice among directed acyclic graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian DAG models that satisfies our assumptions is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let $W$ be an $n \times n$, $n \ge 3$, 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.


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Dan Geiger. David Heckerman. "Parameter priors for directed acyclic graphical models and the characterization of several probability distributions." Ann. Statist. 30 (5) 1412 - 1440, October 2002.


Published: October 2002
First available in Project Euclid: 28 October 2002

zbMATH: 1016.62064
MathSciNet: MR1936324
Digital Object Identifier: 10.1214/aos/1035844981

Primary: 60E05 , 62E10
Secondary: 39B99 , 62A15 , 62C10

Keywords: Bayesian network , directed acyclic graphical model , Dirichlet distribution , Gaussian DAG model , learning , linear regression model , normal distribution , Wishart distribution

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 5 • October 2002
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