In response to everyday queries, humans explicitly signal uncertainty and offer alternative answers when they are unsure. Machine learning models that output calibrated prediction sets through conformal prediction mimic this human behaviour; larger sets signal greater uncertainty while providing alternatives. In this work, we study the usefulness of conformal prediction sets as an aid for human decision making by conducting a pre-registered randomized controlled trial with conformal prediction sets provided to human subjects. With statistical significance, we find that when humans are given conformal prediction sets their accuracy on tasks improves compared to fixed-size prediction sets with the same coverage guarantee. The results show that quantifying model uncertainty with conformal prediction is helpful for human-in-the-loop decision making and human-AI teams.

Conformal prediction is a statistically rigorous method for quantifying uncertainty in models by having them output sets of predictions, with larger sets indicating more uncertainty. However, prediction sets are not inherently actionable; many applications require a single output to act on, not several. To overcome this limitation, prediction sets can be provided to a human who then makes an informed decision. In any such system it is crucial to ensure the fairness of outcomes across protected groups, and researchers have proposed that Equalized Coverage be used as the standard for fairness. By conducting experiments with human participants, we demonstrate that providing prediction sets can lead to disparate impact in decisions. Disquietingly, we find that providing sets that satisfy Equalized Coverage actually increases disparate impact compared to marginal coverage. Instead of equalizing coverage, we propose to equalize set sizes across groups which empirically leads to lower disparate impact.

The classical Frobenius problem is to compute the largest number g not representable as a non-negative integer linear combination of non-negative integers x_1, x_2, ..., x_k, where gcd(x_1, x_2, ..., x_k) = 1. In this paper we consider generalizations of the Frobenius problem to the noncommutative setting of a free monoid. Unlike the commutative case, where the bound on g is quadratic, we are able to show exponential or subexponential behavior for an analogue of g, depending on the particular measure chosen.