Representation of Context-Specific Causal Models with Observational and Interventional Data

We consider the problem of representing causal models that encode context-specific information for discrete data. To represent such models we use a proper subclass of staged tree models which we call CStrees. We show that the context-specific information encoded by a CStree can be equivalently expressed via a collection of DAGs. As not all staged tree models admit this property, CStrees are a subclass that provides a transparent, intuitive and compact representation of context-specific causal information. Model equivalence for CStrees also takes a simpler form than for general staged trees: We provide a characterization of the complete set of asymmetric conditional independence relations encoded by a CStree. As a consequence, we obtain a global Markov property for CStrees which leads to a graphical criterion of model equivalence for CStrees generalizing that of Verma and Pearl for DAG models. In addition, we provide a closed-form formula for the maximum likelihood estimator of a CStree and use it to show that the Bayesian information criterion is a locally consistent score function for this model class. We also give an analogous global Markov property and characterization of model equivalence for general interventions in CStrees. As examples, we apply these results to two real data sets, and examine how BIC-optimal CStrees for each provide a clear and concise representation of the learned context-specific causal structure.