Since the rise of fair machine learning as a critical field of inquiry, many different notions on how to quantify and measure discrimination have been proposed in the literature. Some of these notions, however, were shown to be mutually incompatible. Such findings make it appear that numerous different kinds of fairness exist, thereby making a consensus on the appropriate measure of fairness harder to reach, hindering the applications of these tools in practice. In this paper, we investigate one of these key impossibility results that relates the notions of statistical and predictive parity. Specifically, we derive a new causal decomposition formula for the fairness measures associated with predictive parity, and obtain a novel insight into how this criterion is related to statistical parity through the legal doctrines of disparate treatment, disparate impact, and the notion of business necessity. Our results show that through a more careful causal analysis, the notions of statistical and predictive parity are not really mutually exclusive, but complementary and spanning a spectrum of fairness notions through the concept of business necessity. Finally, we demonstrate the importance of our findings on a real-world example.

翻译：自公平机器学习作为关键研究领域兴起以来，文献中提出了多种关于如何量化和衡量歧视的概念。然而，其中一些概念被证明相互不兼容。这些发现似乎表明存在多种不同的公平性，从而使得就公平性的适当衡量标准达成共识更加困难，阻碍了这些工具在实际中的应用。本文研究了与统计均等和预测均等概念相关的一个关键的不可能结果。具体而言，我们为与预测均等相关的公平性度量推导出一个新的因果分解公式，并获得了关于该标准如何通过异类待遇、异类影响的法律原则以及商业必要性概念而与统计均等相关联的新见解。我们的结果表明，通过更细致的因果分析，统计均等和预测均等概念实际上并非相互排斥，而是通过商业必要性概念互补并涵盖了一系列公平性概念。最后，我们通过一个实际例子展示了我们研究结果的重要性。