The Annals of Statistics

A new prior for discrete DAG models with a restricted set of directions

Hélène Massam and Jacek Wesołowski

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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.

Article information

Ann. Statist., Volume 44, Number 3 (2016), 1010-1037.

Received: April 2015
Revised: September 2015
First available in Project Euclid: 11 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H17: Contingency tables 62F15: Bayesian inference 62E99: None of the above, but in this section

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


Massam, Hélène; Wesołowski, Jacek. A new prior for discrete DAG models with a restricted set of directions. Ann. Statist. 44 (2016), no. 3, 1010--1037. doi:10.1214/15-AOS1396.

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Supplemental materials

  • Proofs and some detailed examples for “A new prior for discrete DAG models with a restricted set of directions”. Supplement A contains proofs and examples. We provide the proof of Lemma 2.1 and give a simple example of $\mathfrak{p}$-perfect ordering of the cliques and vertices as given in (2.1). We also provide the proofs of Theorems 4.1, 5.1, 5.2 and 6.1. We give the details of the derivation of the four $\mathcal{P}$-Dirichlet families in Example 3.1 continued. We also illustrate, with two examples, a possible extension of the $\mathcal{P}$-Dirichlet distribution to arbitrary DAGs through the use of essential graphs.