From Representation Identifiability to Counterfactual Identifiability in Latent Structural Causal Models

This paper investigates whether latent structural causal models (SCMs) learned from factual data can be trusted on counterfactuals, that is, whether latent SCMs that fit the same factual data agree on every counterfactual. To answer this question, we define counterfactual identifiability on latent SCMs, introduce the latent SCM isomorphism that implies it, and establish sufficient assumptions for reaching the isomorphism: an auxiliary variable that modulates the exogenous variables and one perfect intervention on each non-root endogenous variable. We further show that the isomorphism supports causal effect analysis at the counterfactual level, identifying natural direct and indirect effects up to a common scaling. Our theoretical results extend causal representation learning by turning representation identification, in the form of the isomorphism, into a guarantee for counterfactual identification. Finally, we develop FlowSCM, an estimator built on our assumptions and theory, and show on synthetic and image data that it recovers the latent SCM and returns its counterfactuals and causal effects.