Causal regularization was introduced as a stable causal inference strategy in a two-environment setting in \cite{kania2022causal}. We start with observing that causal regularizer can be extended to several shifted environments. 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 \cite{kania2022causal} as well as in \cite{Rot}.

翻译：因果正则化最初在文献《kania2022causal》中作为双环境设置下的一种稳定因果推断策略被提出。我们首先观察到，该因果正则化方法可扩展至多个偏移环境。我们在总体设置下推导出多环境因果正则化器，提出了其插件估计量，并研究了该估计量的测度集中行为。尽管该插件估计量的方差在一般情况下并未良定义，我们转而研究了其条件方差，包括相对于经验数据的自然滤子以及针对特定事件的条件化分析。此外，我们还研究了估计量高阶中心绝对矩的条件期望的推广形式。本文所呈现的结果对于文献《kania2022causal》及《Rot》中的先前设置而言亦具有新颖性。