  Racial and other demographic imputation is necessary for many applications,
especially in auditing disparities and outreach targeting in political
campaigns. The canonical approach is to construct continuous predictions --
e.g., based on name and geography -- and then to $\textit{discretize}$ the
predictions by selecting the most likely class (argmax). We study how this
practice produces $\textit{discretization bias}$. In particular, we show that
argmax labeling, as used by a prominent commercial voter file vendor to impute
race/ethnicity, results in a substantial under-count of African-American
voters, e.g., by 28.2% points in North Carolina. This bias can have substantial
implications in downstream tasks that use such labels.
  We then introduce a $\textit{joint optimization}$ approach -- and a tractable
$\textit{data-driven thresholding}$ heuristic -- that can eliminate this bias,
with negligible individual-level accuracy loss. Finally, we theoretically
analyze discretization bias, show that calibrated continuous models are
insufficient to eliminate it, and that an approach such as ours is necessary.
Broadly, we warn researchers and practitioners against discretizing continuous
demographic predictions without considering downstream consequences.
