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

Assessing Algorithmic Fairness with Unobserved Protected Class Using Data Combination