  In algorithmically fair prediction problems, a standard goal is to ensure the
equality of fairness metrics across multiple overlapping groups simultaneously.
We reconsider this standard fair classification problem using a probabilistic
population analysis, which, in turn, reveals the Bayes-optimal classifier. Our
approach unifies a variety of existing group-fair classification methods and
enables extensions to a wide range of non-decomposable multiclass performance
metrics and fairness measures. The Bayes-optimal classifier further inspires
consistent procedures for algorithmically fair classification with overlapping
groups. On a variety of real datasets, the proposed approach outperforms
baselines in terms of its fairness-performance tradeoff.
