The increasing impact of algorithmic decisions on people's lives compels us to scrutinize their fairness. We consider the use of an auxiliary dataset, such as the US census, to construct models that predict protected classes. We show that a variety of common disparity measures are generally unidentifiable.
The increasing impact of algorithmic decisions on people's lives compels us to scrutinize their fairness and, in particular, the disparate impacts that ostensibly-color-blind algorithms can have on different groups. Examples include credit decisioning, hiring, advertising, criminal justice, personalized medicine, and targeted policymaking, where in some cases legislative or regulatory frameworks for fairness exist and define specific protected classes. In this paper we study a fundamental challenge to assessing disparate impacts in practice: protected class membership is often not observed in the data. This is particularly a problem in lending and healthcare. We consider the use of an auxiliary dataset, such as the US census, to construct models that predict the protected class from proxy variables, such as surname and geolocation. We show that even with such data, a variety of common disparity measures are generally unidentifiable, providing a new perspective on the documented biases of popular proxy-based methods. We provide exact characterizations of the tightest-possible set of all possible true disparities that are consistent with the data (and possibly any assumptions). We further provide optimization-based algorithms for computing and visualizing these sets and statistical tools to assess sampling uncertainty. Together, these enable reliable and robust assessments of disparities -- an important tool when disparity assessment can have far-reaching policy implications. We demonstrate this in two case studies with real data: mortgage lending and personalized medicine dosing.