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**Parameter priors for directed acyclic graphical models and the characterization of several probability distributions.**
*(English)*
Zbl 1016.62064

Summary: 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 \(n\times n\), \(n\geq 3\), 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.

Our analysis is based on the following new characterization of the Wishart distribution: let \(W\) be an \(n\times n\), \(n\geq 3\), 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.

### MSC:

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

05C90 | Applications of graph theory |

60E05 | Probability distributions: general theory |

62C10 | Bayesian problems; characterization of Bayes procedures |

39B99 | Functional equations and inequalities |

05C20 | Directed graphs (digraphs), tournaments |

### Keywords:

Bayesian network; directed acyclic graphical model; Dirichlet distribution; Gaussian DAG model; learning; linear regression model; normal distribution; Wishart distribution### Software:

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\textit{D. Geiger} and \textit{D. Heckerman}, Ann. Stat. 30, No. 5, 1412--1440 (2002; Zbl 1016.62064)

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