Causal Identification under Markov Equivalence: Completeness Results

Causal effect identification is the task of determining whether a causal distribution is computable from the combination of an observational distribution and substantive knowledge about the domain under investigation. One of the most studied versions of this problem assumes that knowledge is articulated in the form of a fully known causal diagram, which is arguably a strong assumption in many settings. In this paper, we relax this requirement and consider that the knowledge is articulated in the form of an equivalence class of causal diagrams, in particular, a partial ancestral graph (PAG). This is attractive because a PAG can be learned directly from data, and the data scientist does not need to commit to a particular, unique diagram. There are different sufficient conditions for identification in PAGs, but none is complete. We derive a complete algorithm for identification given a PAG. This implies that whenever the causal effect is identifiable, the algorithm returns a valid identification expression; alternatively, it will throw a failure condition, which means that the effect is provably not identifiable (unless stronger assumptions are made). We further provide a graphical characterization of non-identifiability of causal effects in PAGs.