Learning by Confusion Locates a Parametrization-Dependent Fisher Information Maximum
Abstract
Learning by confusion (LBC) is a machine-learning method for locating phase transitions without assuming a known order parameter. Its characteristic accuracy curve contains a central maximum that is interpreted as the transition point, but what determines the location and finite-size scaling of this maximum has remained unclear. We derive an exact balance condition for the Bayes-optimal confusion peak and show that, for equilibrium systems with a single swept parameter, the peak closely follows the maximum of the corresponding Fisher information. This immediately gives its finite-size scaling and reveals an unexpected dependence on how the data are sampled: the same configurations arranged on grids uniform in T, ln T, or β=1/T can yield different finite-size pseudocritical trajectories. Tests on the Ising, Blume-Capel, Potts, and Baxter-Wu models confirm the prediction across continuous, tricritical, and first-order transitions. Our results identify the statistical quantity measured by LBC and show that finite-size machine-learning estimates can depend on the parametrization of the training data as well as on the underlying physics.
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BibTeX
@misc{issa2026learning,
title = {{Learning by Confusion Locates a Parametrization-Dependent Fisher Information Maximum}},
author = {George Issa and Richard Scalettar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34769},
eprint = {2610.34769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34769},
}
RIS
TY - UNPB
TI - Learning by Confusion Locates a Parametrization-Dependent Fisher Information Maximum
AU - George Issa
AU - Richard Scalettar
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34769
UR - https://arxiv.org/abs/2610.34769
AB - Learning by confusion (LBC) is a machine-learning method for locating phase transitions without assuming a known order parameter. Its characteristic accuracy curve contains a central maximum that is interpreted as the transition point, but what determines the location and finite-size scaling of this maximum has remained unclear. We derive an exact balance condition for the Bayes-optimal confusion peak and show that, for equilibrium systems with a single swept parameter, the peak closely follows the maximum of the corresponding Fisher information. This immediately gives its finite-size scaling and reveals an unexpected dependence on how the data are sampled: the same configurations arranged on grids uniform in T, ln T, or β=1/T can yield different finite-size pseudocritical trajectories. Tests on the Ising, Blume-Capel, Potts, and Baxter-Wu models confirm the prediction across continuous, tricritical, and first-order transitions. Our results identify the statistical quantity measured by LBC and show that finite-size machine-learning estimates can depend on the parametrization of the training data as well as on the underlying physics.
ER -