Characterization and Greedy Learning of Interventional Markov Equivalence Classes of Directed Acyclic Graphs (1104.2808v2)

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.

• The paper introduces interventional Markov equivalence to refine DAG distinctions using both observational and interventional data.
• The paper presents the GIES algorithm, extending GES to efficiently search interventional equivalence classes with a BIC-based scoring method.
• The paper validates its approach through simulations, showing enhanced accuracy in causal structure learning even with a limited number of interventions.

Characterization and Greedy Learning of Interventional Markov Equivalence Classes of Directed Acyclic Graphs

The paper by Hauser and Bühlmann investigates the concept of interventional Markov equivalence classes for Directed Acyclic Graphs (DAGs) within the context of structure learning with observational and interventional data. The paper extends traditional observational Markov equivalence, providing a new graph theoretical and algorithmic framework to improve the identifiability of causal models with interventions.

Summary of Contributions

1. Interventional Markov Equivalence: The authors introduce the notion of interventional Markov equivalence classes that leverage the conditional independence relations induced by interventions, proposing a graph theoretic criterion that defines a finer partitioning of DAGs. It permits distinguishing between DAGs that are equivalent under observational data but not under interventional data.
2. Interventional Essential Graphs: They establish that an interventional essential graph represents each interventional Markov equivalence class. These graphs, analogous to chain graphs with chordal chain components, allow for understanding and utilizing the enhanced identifiability of causal structures under interventions.
3. Greedy Interventional Equivalence Search (GIES): The paper develops GIES, an algorithm extending the Greedy Equivalence Search (GES) to interventional data. GIES traverses the space of interventional Markov equivalence classes efficiently—starting from an empty graph, it adds and turns arrows to maximize a score function based on BIC, ensuring computational tractability in practical applications.
4. Simulation and Evaluation: The application of these methods is evaluated through simulations showing substantial improvements in structure learning accuracy relative to traditional methods. GIES, in particular, demonstrates robust performance even with a limited number of interventions, suggesting its potential efficacy in real-world applications where extensive experimentation is costly or impractical.

Implications and Future Directions

The proposed methods render DAGs more identifiable by integrating both observational and interventional data, which is pivotal for causal inference and complex system modeling. The ability to obtain tighter equivalence classes through interventional essential graphs signifies a paradigm shift, particularly in fields such as genomics and systems biology, where controlled experiments can provide intricate insights into system dynamics.

The framework allows researchers to explore and refine causal hypotheses at a granularity impossible with observational data alone, thus expanding potential applications in policy-making, medical diagnostics, and econometrics.

From a theoretical standpoint, the consistency of GIES remains a crucial open question, with investigations into consistency conditions and sensitivity analyses for the score functions posing intriguing research avenues.

In conclusion, this paper extends the analytical capabilities for causal discovery with interventional data, providing a foundational basis for further exploration into adaptable and optimized algorithms and frameworks for the broad application spectrum where causal inference plays a vital role. As interventions become more accessible and computational resources advance, the resultant insights could lead to significant breakthroughs across various scientific disciplines.