Minimal enumeration of all possible total effects in a Markov\n equivalence class

In observational studies, when a total causal effect of interest is not\nidentified, the set of all possible effects can be reported instead. This\ntypically occurs when the underlying causal DAG is only known up to a Markov\nequivalence class, or a refinement thereof due to background knowledge. As\nsuch, the class of possible causal DAGs is represented by a maximally oriented\npartially directed acyclic graph (MPDAG), which contains both directed and\nundirected edges. We characterize the minimal additional edge orientations\nrequired to identify a given total effect. A recursive algorithm is then\ndeveloped to enumerate subclasses of DAGs, such that the total effect in each\nsubclass is identified as a distinct functional of the observed distribution.\nThis resolves an issue with existing methods, which often report possible total\neffects with duplicates, namely those that are numerically distinct due to\nsampling variability but are in fact causally identical.\n