  It is widely accepted that biased data leads to biased and thus potentially
unfair models. Therefore, several measures for bias in data and model
predictions have been proposed, as well as bias mitigation techniques whose aim
is to learn models that are fair by design. Despite the myriad of mitigation
techniques developed in the past decade, however, it is still poorly understood
under what circumstances which methods work. Recently, Wick et al. showed, with
experiments on synthetic data, that there exist situations in which bias
mitigation techniques lead to more accurate models when measured on unbiased
data. Nevertheless, in the absence of a thorough mathematical analysis, it
remains unclear which techniques are effective under what circumstances. We
propose to address this problem by establishing relationships between the type
of bias and the effectiveness of a mitigation technique, where we categorize
the mitigation techniques by the bias measure they optimize. In this paper we
illustrate this principle for label and selection bias on the one hand, and
demographic parity and ``We're All Equal'' on the other hand. Our theoretical
analysis allows to explain the results of Wick et al. and we also show that
there are situations where minimizing fairness measures does not result in the
fairest possible distribution.
