Stable mixed graphs

TL;DR

This paper introduces stable mixed graphs, including ribbonless graphs, which model the independence structure of DAGs after marginalisation and conditioning, providing algorithms to generate and relate these graph classes.

Contribution

It defines ribbonless graphs as a new class, extending MC graphs, and develops algorithms for generating and relating these graph classes from DAGs.

Findings

Ribbonless graphs permit the use of the m-separation criterion.

Algorithms are provided to generate RGs, summary, and ancestral graphs from DAGs.

The paper establishes relationships among these graph classes.

Abstract

In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the $m$-separation criterion. These include MC, summary, and ancestral graphs. As a modification of MC graphs, we define the class of ribbonless graphs (RGs) that permits the use of the $m$-separation criterion. RGs contain summary and ancestral graphs as subclasses, and each RG can be generated by a DAG after marginalisation and conditioning. We derive simple algorithms to generate RGs, from given DAGs or RGs, and also to generate summary and ancestral graphs in a simple way by further extension of the RG-generating algorithm. This enables us to develop a parallel theory on these three classes and to study the relationships between them as well as the use of each class.