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June 2016 A new prior for discrete DAG models with a restricted set of directions
Hélène Massam, Jacek Wesołowski
Ann. Statist. 44(3): 1010-1037 (June 2016). DOI: 10.1214/15-AOS1396

Abstract

In this paper, we first develop a new family of conjugate prior distributions for the cell probability parameters of discrete graphical models Markov with respect to a set $\mathcal{P}$ of moral directed acyclic graphs with skeleton a given decomposable graph $G$. This family, which we call the $\mathcal{P}$-Dirichlet, is a generalization of the hyper Dirichlet given in [Ann. Statist. 21 (1993) 1272–1317]: it keeps the directed strong hyper Markov property for every DAG in $\mathcal{P}$ but increases the flexibility in the choice of its parameters, that is, the hyper parameters.

Our second contribution is a characterization of the $\mathcal{P}$-Dirichlet, which yields, as a corollary, a characterization of the hyper Dirichlet and a characterization of the Dirichlet also. Like the characterization of the Dirichlet given in [Ann. Statist. 25 (1997) 1344–1369], our characterization of the $\mathcal{P}$-Dirichlet is based on local and global independence of the probability parameters and also a separability property explicitly defined here but implicitly used in that paper through the choice of two particular DAGs. Another advantage of our approach is that we need not make the assumption of the existence of a positive density function. We use the method of moments for our proofs.

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Hélène Massam. Jacek Wesołowski. "A new prior for discrete DAG models with a restricted set of directions." Ann. Statist. 44 (3) 1010 - 1037, June 2016. https://doi.org/10.1214/15-AOS1396

Information

Received: 1 April 2015; Revised: 1 September 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1341.62153
MathSciNet: MR3485952
Digital Object Identifier: 10.1214/15-AOS1396

Subjects:
Primary: 62E99 , 62F15 , 62H17

Keywords: Bayesian learning , characterization , conjugate priors , directed strong hyper Markov , hyper Dirichlet distribution , local and global independence

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 3 • June 2016
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