Exogenous Isomorphism for Counterfactual Identifiability

This paper investigates ∼ L3 -identifiability, a form of complete counterfactual identifiability within the Pearl Causal Hierarchy (PCH) framework, ensuring that all Structural Causal Models (SCMs) satisfying the given assumptions provide consistent answers to all causal questions. To simplify this problem, we introduce exogenous isomorphism and propose ∼ EI -identifiability, reflecting the strength of model identifiability required for ∼ L3 -identifiability. We explore sufficient assumptions for achieving ∼ EI -identifiability in two special classes of SCMs: Bijective SCMs (BSCMs), based on counterfactual transport, and Triangular Monotonic SCMs (TM-SCMs), which extend ∼ L2 -identifiability. Our results unify and generalize existing theories, providing theoretical guarantees for practical applications. Finally, we leverage neural TM-SCMs to address the consistency problem in counterfactual reasoning, with experiments validating both the effectiveness of our method and the correctness of the theory. Exogenous Isomorphism for Counterfactual Identifiability lence relation between SCMs, denoted as M (1) ∼ φ φ φ M (2) , the problem of causal identification with respect to φ φ φ can be reframed as a problem of ∼ φ φ φ -identifiability. The CHT states that when A contains only low-level knowledge, ∼ φ φ φ -identifiability is often unattainable. Achieving this requires assuming higher-level information. For instance, under the assumptions of known L 1 observational distribution, causal graph, and Markovianity, the causal graph encodes structural constraints on L 2 , enabling the constructed CBN to answer all questions in L 2 . Since the union of all statements φ φ φ ⊆ L 2 equals L 2 , this property is also referred to as ∼ L2 -identifiability. This paper focuses on ∼ L3 -identifiability, which is an enhanced version of ∼ L2 -identifiability, requiring ∼ φ φ φidentifiability for any φ φ φ ⊆ L 3 . If A satisfies ∼ L3identifiability, then by the definition of L 3 -consistency, any 2) . Since L 3 represents the highest level in the PCH and encodes all causal information of the SCM, if M (1) ∼ L3 M (2) , the two models are indistinguishable under any causal statement. Therefore, ∼ L3 -identifiability is the ultimate goal for causal identifiability within the PCH.