  In this paper, we study the problem of fair classification in the presence of
prior probability shifts, where the training set distribution differs from the
test set. This phenomenon can be observed in the yearly records of several
real-world datasets, such as recidivism records and medical expenditure
surveys. If unaccounted for, such shifts can cause the predictions of a
classifier to become unfair towards specific population subgroups. While the
fairness notion called Proportional Equality (PE) accounts for such shifts, a
procedure to ensure PE-fairness was unknown.
  In this work, we propose a method, called CAPE, which provides a
comprehensive solution to the aforementioned problem. CAPE makes novel use of
prevalence estimation techniques, sampling and an ensemble of classifiers to
ensure fair predictions under prior probability shifts. We introduce a metric,
called prevalence difference (PD), which CAPE attempts to minimize in order to
ensure PE-fairness. We theoretically establish that this metric exhibits
several desirable properties.
  We evaluate the efficacy of CAPE via a thorough empirical evaluation on
synthetic datasets. We also compare the performance of CAPE with several
popular fair classifiers on real-world datasets like COMPAS (criminal risk
assessment) and MEPS (medical expenditure panel survey). The results indicate
that CAPE ensures PE-fair predictions, while performing well on other
performance metrics.
