  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 discriminated
against. It is known that ensuring \emph{marginal fairness} for every dimension
independently is not sufficient in general. Due to the exponential number of
subgroups, however, directly measuring intersectional fairness from data is
impossible. In this paper, our primary goal is to understand in detail the
relationship between marginal and intersectional fairness through statistical
analysis. We first identify a set of sufficient conditions under which an exact
relationship can be obtained. Then, we prove bounds (easily computable through
marginal fairness and other meaningful statistical quantities) in
high-probability on intersectional fairness in the general case. Beyond their
descriptive value, we show that these theoretical bounds can be leveraged to
derive a heuristic improving the approximation and bounds of intersectional
fairness by choosing, in a relevant manner, protected attributes for which we
describe intersectional subgroups. Finally, we test the performance of our
approximations and bounds on real and synthetic data-sets.
