Counterfactual inference aims to answer retrospective ''what if'' questions and thus belongs to the most fine-grained type of inference in Pearl's causality ladder. Existing methods for counterfactual inference with continuous outcomes aim at point identification and thus make strong and unnatural assumptions about the underlying structural causal model. In this paper, we relax these assumptions and aim at partial counterfactual identification of continuous outcomes, i.e., when the counterfactual query resides in an ignorance interval with informative bounds. We prove that, in general, the ignorance interval of the counterfactual queries has non-informative bounds, already when functions of structural causal models are continuously differentiable. As a remedy, we propose a novel sensitivity model called Curvature Sensitivity Model. This allows us to obtain informative bounds by bounding the curvature of level sets of the functions. We further show that existing point counterfactual identification methods are special cases of our Curvature Sensitivity Model when the bound of the curvature is set to zero. We then propose an implementation of our Curvature Sensitivity Model in the form of a novel deep generative model, which we call Augmented Pseudo-Invertible Decoder. Our implementation employs (i) residual normalizing flows with (ii) variational augmentations. We empirically demonstrate the effectiveness of our Augmented Pseudo-Invertible Decoder. To the best of our knowledge, ours is the first partial identification model for Markovian structural causal models with continuous outcomes.

翻译：反事实推理旨在回答回顾性“假设”问题，属于Pearl因果阶梯中最细粒度的推断类型。现有针对连续结果的反事实推断方法聚焦于点识别，因此对底层结构因果模型做出了强且不自然的假设。本文放宽了这些假设，旨在实现连续结果的部分反事实识别，即当反事实查询处于具有信息性边界的无知区间时。我们证明，一般而言，当结构因果模型的函数连续可微时，反事实查询的无知区间已具有非信息性边界。作为解决方案，我们提出一种新颖的敏感性模型——曲率敏感性模型。通过限制函数水平集的曲率边界，我们能够获得信息性边界。进一步证明，当曲率边界设为零时，现有的点反事实识别方法是我们曲率敏感性模型的特殊情况。随后，我们提出以新型深度生成模型——增广伪可逆解码器——实现曲率敏感性模型。该实现采用（i）残差归一化流结合（ii）变分增广。我们通过实验验证了增广伪可逆解码器的有效性。据我们所知，这是首个针对连续结果的马尔可夫结构因果模型的部分识别模型。