Learning latent causal models from data has many important applications such as robustness, model extrapolation, and counterfactuals. Most prior theoretic work has focused on full causal discovery (i.e., recovering the true latent variables) but requires strong assumptions such as linearity or fails to have any analysis of the equivalence class of solutions (e.g., IRM). Instead of full causal discovery, we focus on a specific type of causal query called the domain counterfactual, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, we assume domain-specific invertible latent structural causal models and a shared invertible observation function, both of which are less restrictive assumptions than prior theoretic works. Under these assumptions, we define domain counterfactually equivalent models and prove that any model can be transformed into an equivalent model via two invertible functions. This constructive property provides a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG, i.e., all non-intervened variables have non-intervened ancestors. This surprising result suggests that an algorithm that only allows intervention in the last $k$ latent variables may improve model estimation for counterfactuals. In experiments, we enforce the sparse intervention hypothesis via this theoretic result by constraining that the latent SCMs can only differ in the last few causal mechanisms and demonstrate the feasibility of this algorithm in simulated and image-based experiments.

翻译：从数据中学习潜在因果模型在鲁棒性、模型外推及反事实推断等领域具有重要应用。现有理论工作大多关注完整因果发现（即恢复真实潜在变量），但需强线性假设等严格条件，或缺乏对解等价类（如IRM）的系统分析。不同于完整因果发现，本研究聚焦于名为"域反事实"的特定因果查询，该查询旨在假设样本若在不同域（或环境）中生成时会呈现的形态。具体而言，我们假设存在域特异性可逆潜在结构因果模型与共享可逆观测函数，其限制条件较现有理论工作更为宽松。在此假设下，我们定义了域反事实等价模型，并证明任何模型均可通过两个可逆函数转化为等价模型。这一构造性性质为域反事实等价类提供了紧凑的特征刻画。基于此结果，我们证明每个等价类均包含一个模型，其中所有受干预变量在因果有向无环图拓扑排序中位于末端，即所有非受干预变量均具有非受干预祖先。这一反直觉结论表明，仅允许对最后k个潜在变量进行干预的算法可能改进反事实估计。实验部分，我们通过约束潜在结构因果模型仅最后若干因果机制存在差异来实现稀疏干预假设，并在模拟与图像实验中验证了该算法的可行性。