[Paper Review] A Polynomial-Time Algorithm for Deciding Markov Equivalence of Directed Cyclic Graphical Models

Thomas Richardson|arXiv (Cornell University)|Feb 13, 2013

Bayesian Modeling and Causal Inference10 references30 citations

TL;DR

This paper presents a polynomial-time algorithm to determine Markov equivalence between directed cyclic graphical models by establishing necessary and sufficient conditions for equivalence based on d-separation relations. The method extends the concept of d-separation to cyclic graphs and enables efficient equivalence checking, resolving a problem previously solvable only in exponential time.

ABSTRACT

Although the concept of d-separation was originally defined for directed acyclic graphs (see Pearl 1988), there is a natural extension of he concept to directed cyclic graphs. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are Markov equivalent. In other words, when two directed cyclic graphs are Markov equivalent, the set of distributions that satisfy a natural extension of the Global Directed Markov condition (Lauritzen et al. 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs; simply chech all of the d-separation relations in each graph. In this paper I state a theorem that gives necessary and sufficient conditions for the Markov equivalence of two directed cyclic graphs, where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs. Although space prohibits inclusion of correctness proofs, they are fully described in Richardson (1994b).

Motivation & Objective

• To extend the concept of d-separation from acyclic to directed cyclic graphs.
• To identify necessary and sufficient conditions for Markov equivalence in directed cyclic graphs.
• To develop a decision procedure for Markov equivalence that runs in polynomial time.
• To provide a computationally efficient alternative to the exponential-time method of checking all d-separation relations.
• To enable practical use of cyclic graphical models in causal inference and probabilistic reasoning.

Proposed method

• Extends the global Markov property to directed cyclic graphs using a natural generalization of d-separation.
• Defines Markov equivalence as the condition where two directed cyclic graphs encode identical d-separation relations.
• Identifies structural characteristics—such as the same skeleton and v-structures—that are necessary and sufficient for equivalence.
• Employs a polynomial-time check of these structural conditions to determine equivalence.
• Adapts the algorithm to avoid exhaustive enumeration of all d-separation relations.
• Relies on theoretical results from Richardson (1994b), which are referenced but not included in this paper.

Experimental results

Research questions

• RQ1 What conditions must two directed cyclic graphs satisfy to be Markov equivalent?
• RQ2 Can Markov equivalence between directed cyclic graphs be decided in polynomial time?
• RQ3 How can d-separation relations be generalized to apply to cyclic graphs?
• RQ4 What structural features of a directed cyclic graph determine its Markov equivalence class?
• RQ5 Is there a computationally efficient alternative to checking all d-separation relations for equivalence?

Key findings

• The paper establishes necessary and sufficient conditions for Markov equivalence in directed cyclic graphs that can be checked in polynomial time.
• The proposed algorithm avoids the exponential-time enumeration of all d-separation relations.
• Markov equivalence in cyclic graphs is fully characterized by identical skeletons and v-structures, analogous to the acyclic case.
• The method enables efficient identification of equivalent models, which is critical for structure learning in cyclic systems.
• The results extend the applicability of graphical models to cyclic causal systems with tractable equivalence testing.
• The algorithm is derived from theoretical foundations in Richardson (1994b), which are cited as the source of correctness proofs.