We present a necessary and sufficient causal identification formula for maximally oriented partially directed acyclic graphs (MPDAGs) and a recursive algorithm for possible causal effect enumeration when the causal effect is not identified.
MPDAGs as a class of graphs include directed acyclic graphs (DAGs), completed partially directed acyclic graphs (CPDAGs), and CPDAGs with added background knowledge. They represent a type of graph that can be learned from observational data and background knowledge under the assumption of no latent variables. Our identification criterion can be seen as a generalization of the g-formula of Robins (1986) or the truncated factorization formula (Pearl, 2009).
We also characterize the minimal additional edge orientations required to identify a given total effect. A recursive algorithm is 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 result resolves an issue with existing methods, which often report possible total effects with duplicates, namely those numerically distinct due to sampling variability but causally identical.
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