Faster algorithms for Markov equivalence

TL;DR

This paper introduces a new non-parametric criterion for Markov equivalence of maximal ancestral graphs (MAGs) and related structures, along with efficient algorithms for verifying equivalence and converting between graph types.

Contribution

It provides a novel characterization of Markov equivalence using heads and tails, and develops polynomial time algorithms for equivalence verification and graph conversion.

Findings

New non-parametric criterion for MAG equivalence

Polynomial time algorithm for equivalence verification

Algorithm for converting ADMGs and summary graphs to MAGs

Abstract

Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm ($O(ne^{2})$, where $n$ and $e$ are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time ($O(n^{2}e)$). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.