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，文献中提出了许多关于如何量化和衡量歧视的不同概念。然而，这些概念中的一些被证明是相互不兼容的。这些发现使得人们认为存在多种不同的公平类型，从而更难就适当的公平度量达成共识，阻碍了这些工具在实践中的应用。在本文中，我们探讨了其中一个涉及统计均等与预测均等相关概念的关键不可行性结果。具体而言，我们推导了一个新的因果分解公式，用于与预测均等相关的公平度量，并通过不同待遇、不同影响的法律原则以及业务必要性概念，获得了关于该准则如何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，统计均等与预测均等的概念并非真正相互排斥，而是互补的，并通过业务必要性概念涵盖了公平概念的谱系。最后，我们在一个真实世界的例子中展示了我们发现的重要性。