Causal Domain Adaptation: An Information Bottleneck Approach

We address a common causal domain adaptation scenario in which the target variable is observed in the source domain but completely unobserved in the target domain. Our goal is to impute the missing target values in the target domain using the remaining observed variables, under various shifts. We cast this problem as learning a compact representation that is stable across mechanisms: retaining information needed to predict the target while filtering out spurious variation. For linear Gaussian causal models, we derive a closed-form Gaussian Information Bottleneck solution that reduces to a canonical-correlation-style projection and can incorporate DAG-aware structure when desired. For nonlinear or non-Gaussian settings, we propose a Variational Information Bottleneck encoder–predictor that scales to high-dimensional data, can be trained on the source domain, and deployed zero-shot in the target domain. Experiments on synthetic and real datasets show that our method consistently produces accurate imputations, enabling practical deployment in high-dimensional causal models and providing a unified, lightweight toolkit for causal domain adaptation.