Sharp and Stable Label-Free Calibration Certificates from Logical Coherence
Abstract
Query-conditioned probabilistic predictors answer logically related questions one at a time, and labels to check their calibration are often missing. For each restriction on what a bettor's stakes may use, a minimax theorem identifies, for finitely supported laws of the answers, the sharp label-free lower bound on the corresponding calibration residual. On bundles of different events its mean, marginal and joint stages can separate strictly, and a bundle's bound cannot be attributed to one question without assumptions on the others. The sharp marginal bound is discontinuous in the answers, and no label-free lower bound continuous in the Wasserstein distance stays within a constant factor of it. Lipschitz stakes give a stable bound, sharp for a smooth calibration residual and, for every positive Lipschitz constant, positive exactly when the sharp marginal bound is. On i.i.d. streams, projected online gradient ascent at fixed nodes gives fixed-horizon certificates whose shortfall is bounded by grid, learning and statistical terms, and predictable betting gives anytime-valid ones. In a preregistered study of four predictors on two data sets, all seven confirmatory certificates were positive (the eighth pair did not pass the pilot). A certificate shows only that calibration is at least this bad; zero means "not refuted".
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BibTeX
@misc{kato2026sharp,
title = {{Sharp and Stable Label-Free Calibration Certificates from Logical Coherence}},
author = {Takahiro Kato},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21567},
eprint = {2610.21567},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21567},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Sharp and Stable Label-Free Calibration Certificates from Logical Coherence
AU - Takahiro Kato
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21567
UR - https://arxiv.org/abs/2610.21567
AB - Query-conditioned probabilistic predictors answer logically related questions one at a time, and labels to check their calibration are often missing. For each restriction on what a bettor's stakes may use, a minimax theorem identifies, for finitely supported laws of the answers, the sharp label-free lower bound on the corresponding calibration residual. On bundles of different events its mean, marginal and joint stages can separate strictly, and a bundle's bound cannot be attributed to one question without assumptions on the others. The sharp marginal bound is discontinuous in the answers, and no label-free lower bound continuous in the Wasserstein distance stays within a constant factor of it. Lipschitz stakes give a stable bound, sharp for a smooth calibration residual and, for every positive Lipschitz constant, positive exactly when the sharp marginal bound is. On i.i.d. streams, projected online gradient ascent at fixed nodes gives fixed-horizon certificates whose shortfall is bounded by grid, learning and statistical terms, and predictable betting gives anytime-valid ones. In a preregistered study of four predictors on two data sets, all seven confirmatory certificates were positive (the eighth pair did not pass the pilot). A certificate shows only that calibration is at least this bad; zero means "not refuted".
ER -