A Proofs

This is essentially by definition--intervention on Z doesn't change the potential outcomes, so it doesn't change the value of f (X). If f is a counterfactually invariant predictor: 1. Let L be either square error or cross entropy loss. Suppose that the target distribution Q is causally compatible with the training distribution P . Suppose that any of the following conditions hold: 1. the data obeys the anti-causal graph 2. the data obeys the causal-direction graph, there is no confounding (but possibly selection), and the association is purely spurious, Y X | X We begin with the anti-causal case.