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Discrimination in machine learning often arises along multiple dimensions(a.k.a. protected attributes); it is then desirable to ensure\emph{intersectional fairness} -- i.e., that no subgroup is discriminatedagainst. It is known that ensuring \emph{marginal fairness} for every dimensionindependently is not sufficient in general. Due to the exponential number ofsubgroups, however, directly measuring intersectional fairness from data isimpossible. In this paper, our primary goal is to understand in detail therelationship between marginal and intersectional fairness through statisticalanalysis. We first identify a set of sufficient conditions under which an exactrelationship can be obtained. Then, we prove bounds (easily computable throughmarginal fairness and other meaningful statistical quantities) inhigh-probability on intersectional fairness in the general case. Beyond theirdescriptive value, we show that these theoretical bounds can be leveraged toderive a heuristic improving the approximation and bounds of intersectionalfairness by choosing, in a relevant manner, protected attributes for which wedescribe intersectional subgroups. Finally, we test the performance of ourapproximations and bounds on real and synthetic data-sets.

Bounding and Approximating Intersectional Fairness through Marginal Fairness