A Variational Estimator for $L_p$ Calibration Errors

Fuente: arXiv
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Main Authors: Berta, Eugène, Braun, Sacha, Holzmüller, David, Bach, Francis, Jordan, Michael I.
Format: Preprint
Published: 2026
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author Berta, Eugène
Braun, Sacha
Holzmüller, David
Bach, Francis
Jordan, Michael I.
author_facet Berta, Eugène
Braun, Sacha
Holzmüller, David
Bach, Francis
Jordan, Michael I.
contents Calibration$\unicode{x2014}$the problem of ensuring that predicted probabilities align with observed class frequencies$\unicode{x2014}$is a basic desideratum for reliable prediction with machine learning systems. Calibration error is traditionally assessed via a divergence function, using the expected divergence between predictions and empirical frequencies. Accurately estimating this quantity is challenging, especially in the multiclass setting. Here, we show how to extend a recent variational framework for estimating calibration errors beyond divergences induced induced by proper losses, to cover a broad class of calibration errors induced by $L_p$ divergences. Our method can separate over- and under-confidence and, unlike non-variational approaches, avoids overestimation. We provide extensive experiments and integrate our code in the open-source package probmetrics (https://github.com/dholzmueller/probmetrics) for evaluating calibration errors.
format Preprint
id arxiv_https___arxiv_org_abs_2602_24230
institution arXiv
publishDate 2026
record_format arxiv
spellingShingle A Variational Estimator for $L_p$ Calibration Errors
Berta, Eugène
Braun, Sacha
Holzmüller, David
Bach, Francis
Jordan, Michael I.
Machine Learning
Calibration$\unicode{x2014}$the problem of ensuring that predicted probabilities align with observed class frequencies$\unicode{x2014}$is a basic desideratum for reliable prediction with machine learning systems. Calibration error is traditionally assessed via a divergence function, using the expected divergence between predictions and empirical frequencies. Accurately estimating this quantity is challenging, especially in the multiclass setting. Here, we show how to extend a recent variational framework for estimating calibration errors beyond divergences induced induced by proper losses, to cover a broad class of calibration errors induced by $L_p$ divergences. Our method can separate over- and under-confidence and, unlike non-variational approaches, avoids overestimation. We provide extensive experiments and integrate our code in the open-source package probmetrics (https://github.com/dholzmueller/probmetrics) for evaluating calibration errors.
title A Variational Estimator for $L_p$ Calibration Errors
topic Machine Learning
url https://arxiv.org/abs/2602.24230