Threshold tests have recently been proposed as a robust 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 Bayesian latent variable model for which inference is often computationally challenging. Here we develop a method for fitting threshold tests that is more than 75 times faster than the existing approach, reducing computation from hours to minutes. We demonstrate this technique by analyzing 2.7 million police stops of pedestrians in New York City between 2008 and 2012. 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. These discriminant distributions may aid inference in a variety of applications beyond threshold tests.
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