Estimating the causal effects of an intervention from high-dimensional observational data is difficult due to the presence of confounding. The task is often complicated by the fact that we may have a systematic missingness in our data at test time. Our approach uses the information bottleneck to perform a low-dimensional compression of covariates by explicitly considering the relevance of information. Based on the sufficiently reduced covariate, we transfer the relevant information to cases where data is missing at test time, allowing us to reliably and accurately estimate the effects of an intervention, even where data is incomplete. Our results on causal inference benchmarks and a real application for treating sepsis show that our method achieves state-of-the art performance, without sacrificing interpretability.