[Paper Review] Representation of Context-Specific Causal Models with Observational and Interventional Data

Eliana Duarte, Liam Solus|arXiv (Cornell University)|Jan 22, 2021

Bayesian Modeling and Causal Inference4 citations

TL;DR

This paper introduces CStrees, a novel class of causal models that represent context-specific conditional independence using a structured subclass of staged trees, enabling compact, DAG-based graphical representations of context-specific causal relationships. The key contribution is a global Markov property and a graphical criterion for model equivalence—extending to interventional settings—making CStrees the first context-specific model family with a complete characterization of interventional model equivalence.

ABSTRACT

We address the problem of representing context-specific causal models based on both observational and experimental data collected under general (e.g. hard or soft) interventions by introducing a new family of context-specific conditional independence models called CStrees. This family is defined via a novel factorization criterion that allows for a generalization of the factorization property defining general interventional DAG models. We derive a graphical characterization of model equivalence for observational CStrees that extends the Verma and Pearl criterion for DAGs. This characterization is then extended to CStree models under general, context-specific interventions. To obtain these results, we formalize a notion of context-specific intervention that can be incorporated into concise graphical representations of CStree models. We relate CStrees to other context-specific models, showing that the families of DAGs, CStrees, labeled DAGs and staged trees form a strict chain of inclusions. We end with an application of interventional CStree models to a real data set, revealing the context-specific nature of the data dependence structure and the soft, interventional perturbations.

Motivation & Objective

• To address the limitation of standard DAGs in encoding context-specific conditional independence (CSI) relations that hold only under specific conditions.
• To develop a model class that retains the interpretability of DAGs while capturing richer CSI structures than Bayesian multinets.
• To provide a global Markov property and a graphical criterion for model equivalence in both observational and interventional settings.
• To establish a closed-form maximum likelihood estimator and prove local consistency of the BIC score for model selection.
• To demonstrate that CStrees offer a compact, interpretable, and statistically consistent alternative to general staged trees and CEGs for causal discovery with discrete data.

Proposed method

• Define CStrees as a proper subclass of staged tree models where context-specific independence is encoded via a sequence of DAGs, one per context of the conditioning variables.
• Prove that CStrees admit a global Markov property, generalizing the d-separation criterion of DAGs to context-specific settings.
• Establish a graphical criterion for model equivalence by comparing skeletons and v-structures across context-specific DAGs in the sequence.
• Extend the equivalence criterion to interventional models by analyzing the effect of interventions on context-specific DAGs and their structural invariants.
• Derive a closed-form maximum likelihood estimator for CStrees based on context-specific probability parameters.
• Demonstrate that the Bayesian Information Criterion (BIC) is locally consistent for CStrees by showing it asymptotically favors the true model structure.

Experimental results

Research questions

• RQ1 Can context-specific conditional independence relations be represented in a way that is both compact and interpretable as a DAG?
• RQ2 Does a global Markov property exist for context-specific causal models that generalizes the d-separation criterion of standard DAGs?
• RQ3 Can a graphical criterion for model equivalence be established for interventional data in context-specific models?
• RQ4 Is the BIC score a locally consistent score function for CStrees, enabling reliable model selection?
• RQ5 How does the predictive performance of CStrees compare to general staged trees in real and simulated data?

Key findings

• CStrees provide a compact, DAG-based representation of context-specific causal models, with each context corresponding to a distinct DAG that captures context-specific dependencies.
• The global Markov property for CStrees generalizes the d-separation criterion of DAGs, enabling a graphical characterization of conditional independence in context-specific settings.
• Model equivalence for CStrees is characterized by identical skeletons and v-structures across all context-specific DAGs, extending Verma and Pearl's criterion to context-specific models.
• The interventional model equivalence criterion is fully characterized by comparing context-specific DAGs under interventions, making CStrees the first family of context-specific models with such a characterization.
• A closed-form maximum likelihood estimator is derived for CStrees, enabling efficient parameter estimation from observational and interventional data.
• The BIC score is proven to be locally consistent for CStrees, supporting its use in score-based causal discovery algorithms without significant loss in predictive accuracy compared to general staged trees.