  Most current domain adaptation methods address either covariate shift or
label shift, but are not applicable where they occur simultaneously and are
confounded with each other. Domain adaptation approaches which do account for
such confounding are designed to adapt covariates to optimally predict a
particular label whose shift is confounded with covariate shift. In this paper,
we instead seek to achieve general-purpose data backwards compatibility. This
would allow the adapted covariates to be used for a variety of downstream
problems, including on pre-existing prediction models and on data analytics
tasks. To do this we consider a special case of generalized label shift (GLS),
which we call confounded shift. We present a novel framework for this problem,
based on minimizing the expected divergence between the source and target
conditional distributions, conditioning on possible confounders. Within this
framework, we propose using the reverse Kullback-Leibler divergence,
demonstrating the use of parametric and nonparametric Gaussian estimators of
the conditional distribution. We also propose using the Maximum Mean
Discrepancy (MMD), introducing a dynamic strategy for choosing the kernel
bandwidth, which is applicable even outside the confounded shift setting.
Finally, we demonstrate our approach on synthetic and real datasets.
