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Answering counterfactual queries has many important applications such asknowledge discovery and explainability, but is challenging when causalvariables are unobserved and we only see a projection onto an observationspace, for instance, image pixels. One approach is to recover the latentStructural Causal Model (SCM), but this typically needs unrealisticassumptions, such as linearity of the causal mechanisms. Another approach is touse na\"ive ML approximations, such as generative models, to generatecounterfactual samples; however, these lack guarantees of accuracy. In thiswork, we strive to strike a balance between practicality and theoreticalguarantees by focusing on a specific type of causal query called domaincounterfactuals, which hypothesizes what a sample would have looked like if ithad been generated in a different domain (or environment). Concretely, by onlyassuming invertibility, sparse domain interventions and access to observationaldata from different domains, we aim to improve domain counterfactual estimationboth theoretically and practically with less restrictive assumptions. We definedomain counterfactually equivalent models and prove necessary and sufficientproperties for equivalent models that provide a tight characterization of thedomain counterfactual equivalence classes. Building upon this result, we provethat every equivalence class contains a model where all intervened variablesare at the end when topologically sorted by the causal DAG. This surprisingresult suggests that a model design that only allows intervention in the last$k$ latent variables may improve model estimation for counterfactuals. We thentest this model design on extensive simulated and image-based experiments whichshow the sparse canonical model indeed improves counterfactual estimation overbaseline non-sparse models.

Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models