Causal regularization was introduced as a stable causal inference strategy in a two-environment setting in Kania and Wit [2022] for linear structural equations models (SEMs). We start with observing that causal regularizer can be extended to several shifted environments and non-linear SEMs. We derive the multi-environment casual regularizer in the population setting. We propose its plug-in estimator, and study its concentration in measure behavior. Although the variance of the plug-in estimator is not well-defined in general, we instead study its conditional variance both with respect to a natural filtration of the empirical as well as conditioning with respect to certain events. We also study generalizations where we consider conditional expectations of higher central absolute moments of the estimator. The results presented here are also new in the prior setting of Kania and Wit [2022] as well as in Rothenhausler et al. [2021].

翻译：因果正则化最初由Kania与Wit [2022]在线性结构方程模型（SEMs）的双环境设置中提出，作为一种稳定的因果推断策略。我们首先发现因果正则化可扩展至多个偏移环境及非线性SEMs。我们在总体设置中推导了多环境因果正则化形式，提出其插件估计量，并研究了该估计量的测度集中性行为。尽管该插件估计量的方差在一般情况下并非良定义，我们转而研究其条件方差——既考虑了基于经验自然过滤的条件，也考虑了基于特定事件的条件化。此外，我们探讨了估计量高阶中心绝对矩的条件期望泛化形式。本文所呈现的结果对于Kania与Wit [2022]及Rothenhausler等 [2021]的前期设置均具有新颖性。