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：反事实推断旨在回答回顾性"如果会怎样"的问题，因此属于珀尔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均着眼于点识别，因而对底层结构因果模型做出了强烈且非自然的假设。本文放宽了这些假设，致力于实现连续结果的部分反事实识别——即当反事实查询位于具有信息性边界的无知区间时。我们证明，当结构因果模型的函数满足连续可微条件时，反事实查询的无知区间通常具有非信息性边界。作为解决方案，我们提出一种称为曲率敏感性模型的新型敏感性模型，通过约束函数水平集的曲率来获得信息性边界。我们进一步证明，当曲率边界设为零时，现有反事实点识别方法均为曲率敏感性模型的特例。随后，我们以新型深度生成模型——增强型伪可逆解码器的形式，提出了曲率敏感性模型的实现方案。该实现采用：(i)残差归一化流结合(ii)变分增强技术。我们通过实验证明了增强型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马尔可夫结构因果模型的部分识别模型。