Causal Robustness from Synthetic Environments

We study robust prediction when the training data come from a single observational distribution. Causality-oriented methods for robust prediction usually use heterogeneity across observed environments both to learn an invariant prediction function and to determine the perturbations against which it is robust. We show that both can be recovered from one distribution under a structural additive-noise mechanism for the response. Under the structural model considered here, the causal prediction function has constant conditional mean squared error given its prediction. We generate synthetic environments by reweighting the covariate distribution along the prediction and use a Cramér--von Mises criterion to measure departures from this invariance. The resulting penalised risk admits a regularised worst-case representation, linking the criterion to robustness against covariate shifts induced by reweighting observations according to their predicted values. Under an additional orthogonality condition, we identify the causal prediction function as the unique risk minimiser subject to this invariance restriction. We establish consistency and an explicit rate, derive a goodness-of-fit test for the invariance restriction, and characterise the population target under misspecification. With multiple observed environments, the construction applies environment by environment and can be combined with Distributional Robustness via Invariant Gradients (DRIG) to control two complementary classes of shifts.