  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.

Machine Learning;Methodology