Markov Properties for Graphical Models with Cycles and Latent Variables

TL;DR

This paper introduces HEDGes, a new class of directed graphs with hyperedges, to model probabilistic graphical models with cycles and latent variables, and analyzes their Markov properties and relationships.

Contribution

It generalizes existing models by combining features of mDAGs and DMGs into HEDGes, and studies their Markov properties and logical relations.

Findings

Markov properties for HEDGes are generally not equivalent with cycles or hyperedges.

HEDGes unify and extend mDAGs and DMGs models.

The paper clarifies logical relations among different Markov properties.

Abstract

We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that can model cycles. We define and analyse several different Markov properties that relate the graphical structure of a HEDG with a probability distribution on a corresponding product space over the set of nodes, for example factorization properties, structural equations properties, ordered/local/global Markov properties, and marginal versions of these. The various Markov properties for HEDGes are in general not equivalent to each other when cycles or hyperedges are present, in contrast with the simpler case of directed acyclic graphical (DAG) models (also known as Bayesian networks). We show how the Markov properties for HEDGes - and thus the corresponding graphical Markov models - are logically related to each other.