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Spurious correlations allow flexible models to predict well during training but poorly on related test populations. Recent work has shown that models that satisfy particular independencies involving correlation-inducing nuisance variables have guarantees on their test performance. Enforcing such independencies requires nuisances to be observed during training. However, nuisances, such as demographics or image background labels, are often missing. Enforcing independence on just the observed data does not imply independence on the entire population. Here we derive MMD estimators used for invariance objectives under missing nuisances. On simulations and clinical data, optimizing through these estimates achieves test performance similar to using estimators that make use of the full data.

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