Markov properties for mixed graphs

TL;DR

This paper introduces loopless mixed graphs to unify Markov properties across various graphical models, demonstrating that independence models induced by m-separation are compositional graphoids and establishing equivalence of Markov properties for maximal ribbonless graphs.

Contribution

It unifies the Markov theory for different graph types using loopless mixed graphs and proves key properties for ribbonless graphs, including maximality and Markov property equivalences.

Findings

Independence models induced by m-separation are compositional graphoids.

Global and pairwise Markov properties are equivalent for maximal ribbonless graphs.

The framework includes undirected, bidirected, and directed acyclic graphs as special cases.

Abstract

In this paper, we unify the Markov theory of a variety of different types of graphs used in graphical Markov models by introducing the class of loopless mixed graphs, and show that all independence models induced by $m$-separation on such graphs are compositional graphoids. We focus in particular on the subclass of ribbonless graphs which as special cases include undirected graphs, bidirected graphs, and directed acyclic graphs, as well as ancestral graphs and summary graphs. We define maximality of such graphs as well as a pairwise and a global Markov property. We prove that the global and pairwise Markov properties of a maximal ribbonless graph are equivalent for any independence model that is a compositional graphoid.