We mathematically compare three competing definitions of group-level nondiscrimination: demographic parity, equalized odds, and calibration. Using the theoretical framework of Friedler et al., we study the properties of each definition under various worldviews, which are assumptions about how, if at all, the observed data is biased. We prove that different worldviews call for different definitions of fairness, and we specify when it is appropriate to use demographic parity and equalized odds. In addition, we argue that calibration is unsuitable for the purpose of ensuring nondiscrimination. Finally, we define a worldview that is more realistic than the previously considered ones, and we introduce a new notion of fairness that is suitable for this worldview.
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