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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].
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