  We propose a method for learning linear models whose predictive performance
is robust to causal interventions on unobserved variables, when noisy proxies
of those variables are available. Our approach takes the form of a
regularization term that trades off between in-distribution performance and
robustness to interventions. Under the assumption of a linear structural causal
model, we show that a single proxy can be used to create estimators that are
prediction optimal under interventions of bounded strength. This strength
depends on the magnitude of the measurement noise in the proxy, which is, in
general, not identifiable. In the case of two proxy variables, we propose a
modified estimator that is prediction optimal under interventions up to a known
strength. We further show how to extend these estimators to scenarios where
additional information about the "test time" intervention is available during
training. We evaluate our theoretical findings in synthetic experiments and
using real data of hourly pollution levels across several cities in China.
