Markov equivalence of marginalized local independence graphs

TL;DR

This paper introduces a new class of directed mixed graphs with μ-separation to represent asymmetric local independence relations, extending graphical models to be closed under marginalization and enabling efficient identification of equivalence classes.

Contribution

It develops the theory of directed mixed graphs with μ-separation, generalizing local independence representations and characterizing Markov equivalence classes with maximal elements.

Findings

Directed mixed graphs with μ-separation form a closed class under marginalization.

Every Markov equivalence class has a maximal element constructed from independence relations.

The directed mixed equivalence graph encodes all identifiable edge information.

Abstract

Symmetric independence relations are often studied using graphical representations. Ancestral graphs or acyclic directed mixed graphs with $m$-separation provide classes of symmetric graphical independence models that are closed under marginalization. Asymmetric independence relations appear naturally for multivariate stochastic processes, for instance in terms of local independence. However, no class of graphs representing such asymmetric independence relations, which is also closed under marginalization, has been developed. We develop the theory of directed mixed graphs with $\mu$-separation and show that this provides a graphical independence model class which is closed under marginalization and which generalizes previously considered graphical representations of local independence. For statistical applications, it is pivotal to characterize graphs that induce the same independence relations as such a Markov equivalence class of graphs is the object that is ultimately identifiable from observational data. Our main result is that for directed mixed graphs with $\mu$-separation each Markov equivalence class contains a maximal element which can be constructed from the independence relations alone. Moreover, we introduce the directed mixed equivalence graph as the maximal graph with edge markings. This graph encodes all the information about the edges that is identifiable from the independence relations, and furthermore it can be computed efficiently from the maximal graph.