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.

本文提出了一种新的敏感度模型 —— 曲率敏感度模型，与传统方法相比，该模型可以实现连续结果的部分反事实证明，并且可以通过对函数的级集进行曲率限制来获得信息，同时还提出了一种 Deep generative model 模型，并通过实验证明了该方法的可行性。