Revisiting the General Identifiability Problem
This work addresses a foundational issue in causal inference for researchers, though it is incremental as it builds on and corrects prior definitions and algorithms.
The paper tackles the general identifiability problem in causal inference by identifying a missing positivity assumption in prior work, which renders existing methods unsound, and proposes a new algorithm that is sound and complete under this assumption, connecting it to classical identifiability through decomposition.
We revisit the problem of general identifiability originally introduced in [Lee et al., 2019] for causal inference and note that it is necessary to add positivity assumption of observational distribution to the original definition of the problem. We show that without such an assumption the rules of do-calculus and consequently the proposed algorithm in [Lee et al., 2019] are not sound. Moreover, adding the assumption will cause the completeness proof in [Lee et al., 2019] to fail. Under positivity assumption, we present a new algorithm that is provably both sound and complete. A nice property of this new algorithm is that it establishes a connection between general identifiability and classical identifiability by Pearl [1995] through decomposing the general identifiability problem into a series of classical identifiability sub-problems.