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\cite{kania2022causal}中被提出，作为双环境场景下的一种稳定因果推断策略。我们首先观察到，因果正则化可以扩展到多个偏移环境。我们在总体设定下推导出多环境因果正则化项，并提出了其插件估计量，同时研究了该估计量的测度集中行为。尽管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——既考虑相对于经验自然滤子的条件，也考虑相对于特定事件的条件。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。本文所呈现的结果在先前文献\cite{kania2022causal}和\cite{Rot}的设定下同样具有新颖性。