Out-of-distribution generalization under random, dense distributional shifts

Many existing approaches for estimating parameters in settings with distributional shifts operate under an invariance assumption. For example, under covariate shift, it is assumed that $p(y|x)$ remains invariant. We refer to such distribution shifts as sparse, since they may be substantial but affect only a part of the data generating system. In contrast, in various real-world settings, shifts might be dense. More specifically, these dense distributional shifts may arise through numerous small and random changes in the population and environment. First, we discuss empirical evidence for such random dense distributional shifts. Then, we develop tools to infer parameters and make predictions for partially observed, shifted distributions. Finally, we apply the framework to several real-world datasets and discuss diagnostics to evaluate the fit of the distributional uncertainty model.