Markov Equivalences for Subclasses of Loopless Mixed Graphs

TL;DR

This paper investigates Markov equivalences within and between subclasses of loopless mixed graphs, especially focusing on maximal ancestral graphs and their subclasses, providing new theoretical insights and algorithms.

Contribution

It introduces novel results on representational Markov equivalence and algorithms for subclasses of loopless mixed graphs, including maximal ancestral graphs.

Findings

Characterization of internal and external Markov equivalences

Algorithms for generating Markov equivalent graphs

New results on representational Markov equivalence

Abstract

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclasses, for representational Markov equivalence, which is the possibility of a graph from a subclass being Markov equivalent to a graph from another subclass, and finding algorithms to generate a graph from a certain subclass that is Markov equivalent to a given graph. We particularly focus on the class of maximal ancestral graphs and its subclasses, namely regression graphs, bidirected graphs, undirected graphs, and directed acyclic graphs, and present novel results for representational Markov equivalence and algorithms.