Assessing algorithmic fairness with unobserved protected class using data combination

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, or performance disparities in general, 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, that includes protected class labels but not decisions or outcomes. We show that a variety of common disparity measures are generally unidentifiable aside for some unrealistic cases, providing a new perspective on the documented biases of popular proxy-based methods. We provide exact characterizations of the sharpest-possible partial identification set of disparities either under no assumptions or when we incorporate mild smoothness constraints. We further provide optimization-based algorithms for computing and visualizing these sets of simultaneously achievable pairwise disparties for assessing disparities that arise between multiple groups, which enables reliable and robust assessments - 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.