We make two contributions to the problem of estimating the L1 calibration error of a binary classifier from a finite dataset. First, we provide an upper bound for any classifier where the calibration function has bounded variation. Second, we provide a method of modifying any classifier so that its calibration error can be upper bounded efficiently without significantly impacting classifier performance and without any restrictive assumptions. All our results are non-asymptotic and distribution-free. We conclude by providing advice on how to measure calibration error in practice. Our methods yield practical procedures that can be run on real-world datasets with modest overhead.
@article{arxiv.2512.13872,
title = {Measuring Uncertainty Calibration},
author = {Kamil Ciosek and Nicolò Felicioni and Sina Ghiassian and Juan Elenter Litwin and Francesco Tonolini and David Gustafsson and Eva Garcia-Martin and Carmen Barcena Gonzalez and Raphaëlle Bertrand-Lalo},
journal= {arXiv preprint arXiv:2512.13872},
year = {2026}
}