Markovian acyclic directed mixed graphs for discrete data

TL;DR

This paper introduces Markovian acyclic directed mixed graphs (ADMGs) for discrete data, providing a factorization criterion, parameterization, and demonstrating that these models form smooth curved exponential families.

Contribution

It generalizes the recursive factorization for DAGs to ADMGs, offering a new parameterization and smoothness results for discrete Markovian models.

Findings

Provides a factorization criterion for ADMGs

Offers a parameterization in terms of simple conditional probabilities

Shows models are smooth curved exponential families

Abstract

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.