  Increasing concerns about disparate effects of AI have motivated a great deal
of work on fair machine learning. Existing works mainly focus on independence-
and separation-based measures (e.g., demographic parity, equality of
opportunity, equalized odds), while sufficiency-based measures such as
predictive parity are much less studied. This paper considers predictive
parity, which requires equalizing the probability of success given a positive
prediction among different protected groups. We prove that, if the overall
performances of different groups vary only moderately, all fair Bayes-optimal
classifiers under predictive parity are group-wise thresholding rules. Perhaps
surprisingly, this may not hold if group performance levels vary widely; in
this case we find that predictive parity among protected groups may lead to
within-group unfairness. We then propose an algorithm we call FairBayes-DPP,
aiming to ensure predictive parity when our condition is satisfied.
FairBayes-DPP is an adaptive thresholding algorithm that aims to achieve
predictive parity, while also seeking to maximize test accuracy. We provide
supporting experiments conducted on synthetic and empirical data.
