[Paper Review] Characterization and Greedy Learning of Interventional Markov Equivalence Classes of Directed Acyclic Graphs

Alain Hauser, Peter Bühlmann|arXiv (Cornell University)|Apr 14, 2011

Bayesian Modeling and Causal Inference38 references240 citations

TL;DR

This paper introduces interventional Markov equivalence classes for directed acyclic graphs (DAGs), extending observational Markov equivalence to incorporate multiple intervention experiments. It proposes a graph-theoretic representation called the interventional essential graph and generalizes the Greedy Equivalence Search (GES) algorithm to Greedy Interventional Equivalence Search (GIES), significantly improving causal structure identifiability from combined observational and interventional data.

ABSTRACT

The investigation of directed acyclic graphs (DAGs) encoding the same Markov property, that is the same conditional independence relations of multivariate observational distributions, has a long tradition; many algorithms exist for model selection and structure learning in Markov equivalence classes. In this paper, we extend the notion of Markov equivalence of DAGs to the case of interventional distributions arising from multiple intervention experiments. We show that under reasonable assumptions on the intervention experiments, interventional Markov equivalence defines a finer partitioning of DAGs than observational Markov equivalence and hence improves the identifiability of causal models. We give a graph theoretic criterion for two DAGs being Markov equivalent under interventions and show that each interventional Markov equivalence class can, analogously to the observational case, be uniquely represented by a chain graph called interventional essential graph (also known as CPDAG in the observational case). These are key insights for deriving a generalization of the Greedy Equivalence Search algorithm aimed at structure learning from interventional data. This new algorithm is evaluated in a simulation study.

Motivation & Objective

• To extend the concept of Markov equivalence in DAGs to include interventional data, enabling finer partitioning of causal models.
• To develop a graph-theoretic representation—interventional essential graphs—that uniquely characterizes interventional Markov equivalence classes.
• To generalize the Greedy Equivalence Search (GES) algorithm for structure learning under interventions, resulting in the Greedy Interventional Equivalence Search (GIES) algorithm.
• To improve the identifiability of causal structures by leveraging both observational and interventional distributions in a unified framework.

Proposed method

• Proposes a graph-theoretic criterion to determine when two DAGs are Markov equivalent under a given set of interventions.
• Introduces interventional essential graphs as a unique, complete representation of interventional Markov equivalence classes, analogous to CPDAGs in the observational case.
• Defines conditions under which interventions refine the partitioning of DAGs beyond observational Markov equivalence, enhancing causal identifiability.
• Develops algorithmic operations to efficiently traverse the space of interventional essential graphs.
• Generalizes the Greedy Equivalence Search (GES) algorithm to handle interventional data, resulting in the Greedy Interventional Equivalence Search (GIES) algorithm.
• Validates the GIES algorithm through a simulation study, demonstrating improved performance in structure learning under interventions.

Experimental results

Research questions

• RQ1 How can Markov equivalence between DAGs be extended to include interventional data, and what conditions ensure that interventions refine the equivalence classes?
• RQ2 What graph-theoretic structure uniquely represents interventional Markov equivalence classes, and what properties does it satisfy?
• RQ3 How can the Greedy Equivalence Search (GES) algorithm be adapted to learn causal structures from both observational and interventional data?
• RQ4 To what extent do interventions improve the identifiability of causal models compared to observational data alone?
• RQ5 What algorithmic operations enable efficient search over interventional essential graphs for structure learning?

Key findings

• Interventional Markov equivalence defines a strictly finer partitioning of DAGs than observational Markov equivalence, thereby improving causal model identifiability under reasonable intervention assumptions.
• Each interventional Markov equivalence class can be uniquely represented by an interventional essential graph, which generalizes the CPDAG structure to the interventional setting.
• The Greedy Interventional Equivalence Search (GIES) algorithm successfully generalizes GES to interventional data, enabling regularized maximum likelihood estimation in this context.
• The GIES algorithm demonstrates improved performance in structure learning on simulated data, particularly in resolving edge orientations that remain ambiguous in observational data.
• Theoretical results, including the existence of DAGs satisfying specific parent and neighbor constraints under interventions, are established via graph-theoretic constructions and cycle-avoidance arguments.
• The implementation of GIES is planned for integration into the R package pcalg, with a prerelease version available upon request.