The Annals of Statistics

Markovian acyclic directed mixed graphs for discrete data

Robin J. Evans and Thomas S. Richardson

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Acyclic directed mixed graphs (ADMGs) are graphs that contain directed ($\rightarrow$) and bidirected ($\leftrightarrow$) edges, subject to the constraint that there are no cycles of directed edges. Such graphs may be used to represent the conditional independence structure induced by a DAG model containing hidden variables on its observed margin. The Markovian model associated with an ADMG is simply the set of distributions obeying the global Markov property, given via a simple path criterion (m-separation). We first present a factorization criterion characterizing the Markovian model that generalizes the well-known recursive factorization for DAGs. For the case of finite discrete random variables, we also provide a parameterization of the model in terms of simple conditional probabilities, and characterize its variation dependence. We show that the induced models are smooth. Consequently, Markovian ADMG models for discrete variables are curved exponential families of distributions.

Article information

Ann. Statist., Volume 42, Number 4 (2014), 1452-1482.

First available in Project Euclid: 7 August 2014

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Zentralblatt MATH identifier

Primary: 62M45: Neural nets and related approaches

Acyclic directed mixed graph curved exponential family conditional independence graphical model m-separation parameterization


Evans, Robin J.; Richardson, Thomas S. Markovian acyclic directed mixed graphs for discrete data. Ann. Statist. 42 (2014), no. 4, 1452--1482. doi:10.1214/14-AOS1206.

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