  The task of distribution generalization concerns making reliable prediction
of a response in unseen environments. The structural causal models are shown to
be useful to model distribution changes through intervention. Motivated by the
fundamental invariance principle, it is often assumed that the conditional
distribution of the response given its predictors remains the same across
environments. However, this assumption might be violated in practical settings
when the response is intervened. In this work, we investigate a class of model
with an intervened response. We identify a novel form of invariance by
incorporating the estimates of certain features as additional predictors.
Effectively, we show this invariance is equivalent to having a deterministic
linear matching that makes the generalization possible. We provide an explicit
characterization of the linear matching and present our simulation results
under various intervention settings.
