  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.
