Abstract

We study a common domain adaptation setting in causal systems with local causal knowledge: the target variable is observed in the source domain but is entirely missing in the target domain, and the conditional mechanism of the target given its Markov blanket is assumed stable across domains. We aim to impute the target variable in the target domain from the remaining observed variables under various shifts. Our central transfer mechanism is structural: restricting the predictor to the Markov blanket of the target screens off shift-prone non-blanket variation and yields zero-shot transfer under blanket invariance. On top of this restriction, we frame estimation as learning a compact, mechanism-stable representation, and we instantiate it with the Information Bottleneck (IB) as a principled compression and regularization mechanism. For linear Gaussian causal models, we derive a closed-form Gaussian Information Bottleneck (GIB) solution that reduces to a canonical correlation analysis (CCA)–style projection and is provably lossless relative to using all non-target variables; in this well-specified regime, ordinary least squares on the blanket is already near-optimal, so the value of IB is regularization rather than accuracy gains. For nonlinear or non-Gaussian data, where no closed-form conditional estimator is available, we introduce a Variational Information Bottleneck (VIB) encoder–predictor that scales to high dimensions and can be trained on source data and deployed zero-shot to the target domain. Across synthetic and real datasets, our approach consistently attains accurate imputations, supporting practical use in high-dimensional causal models and furnishing a unified, lightweight toolkit for causal domain adaptation.