  Threshold tests have recently been proposed as a useful method for detecting
bias in lending, hiring, and policing decisions. For example, in the case of
credit extensions, these tests aim to estimate the bar for granting loans to
white and minority applicants, with a higher inferred threshold for minorities
indicative of discrimination. This technique, however, requires fitting a
complex Bayesian latent variable model for which inference is often
computationally challenging. Here we develop a method for fitting threshold
tests that is two orders of magnitude faster than the existing approach,
reducing computation from hours to minutes. To achieve these performance gains,
we introduce and analyze a flexible family of probability distributions on the
interval [0, 1] -- which we call discriminant distributions -- that is
computationally efficient to work with. We demonstrate our technique by
analyzing 2.7 million police stops of pedestrians in New York City.
