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