PAG2ADMG: An Algorithm for the Complete Causal Enumeration of a Markov Equivalence Class

Causal graphs, such as directed acyclic graphs (DAGs) and partial ancestral graphs (PAGs), represent causal relationships among variables in a model. Methods exist for learning DAGs and PAGs from data and for converting DAGs to PAGs. However, these methods are significantly limited in that they only output a single causal graph consistent with the independencies and dependencies (the Markov equivalence class $M$) estimated from the data. This is problematic and insufficient because many distinct graphs may be consistent with $M$. A data modeler may wish to select among these numerous consistent graphs using domain knowledge or other model selection algorithms. Enumeration of the set of consistent graphs is the bottleneck. In this paper, we present a method that makes this desired enumeration possible. We introduce PAG2ADMG, the first algorithm for enumerating all causal graphs consistent with $M$. PAG2ADMG converts a given PAG into the complete set of acyclic directed mixed graphs (ADMGs) consistent with $M$. We prove the correctness of the approach and demonstrate its efficiency relative to brute-force enumeration.