---
title: "The Calibration Channel Determines the Bayes-Error Proxy: An Exact Law for Temperature-Induced Distortion"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2607.18162v1"
permalink: "https://feed.astralor.com/items/itm_458a85b6440e90a75a3acfd3"
published: "2026-07-20T17:07:26.000Z"
score: 76
channel: "ai"
tags: ["arXiv","Calibration","Temperature Scaling","Bayes-Error Proxy"]
---

# The Calibration Channel Determines the Bayes-Error Proxy: An Exact Law for Temperature-Induced Distortion

## AI commentary

> This paper offers a beautiful mathematical tool that directly expresses the effect of temperature scaling on the error proxy as an identity involving the margin distribution, making the seemingly arbitrary distortion completely predictable. If you work on model calibration or reliability evaluation, especially with soft-label methods, this paper is worth reading: it reminds you that any proxy value must be interpreted within the context of the calibration method, otherwise it can be misleading. Particularly important for researchers in Bayes-error estimation and calibration theory.

## At a glance

This paper precisely characterizes how temperature scaling—the most widely used post-hoc calibration method—distorts the soft-label Bayes-error proxy. The authors prove a model-free identity reducing the temperature-scaled proxy to the classifier's margin distribution, yielding strict monotonicity in temperature and a continuous bijection from temperature to (0,1/2), meaning a fixed classifier can report any proxy value. On CIFAR-10, Fashion-MNIST, and SVHN, the proxy varies 56x to 980x at constant test error, and the calibration temperature minimizing expected calibration error does not coincide with any stable proxy value. The results reinforce that a proxy value is meaningful only together with the mechanism that produced its probabilities.

## Key points

- Proved strict monotonicity of the temperature-scaled Bayes-error proxy in temperature and a continuous bijection between temperature axis and (0,1/2)
- Derived a two-parameter closed form for the proxy-versus-temperature curve under a Gaussian logit model
- Proxy varies 56x to 980x at constant test error across 8 binary tasks
- Temperature minimizing expected calibration error does not coincide with any stable proxy value

## Full text

The soft-label Bayes-error estimator beta(z) = E[min(z, 1-z)] of Ishida et al. estimates the irreducible error of a binary task directly from probability-valued labels. Recent work by Ushio et al. showed that this estimator is fragile when the probabilities are not the true posterior: even perfectly calibrated soft labels can yield a substantially inaccurate estimate, and they propose isotonic calibration as a consistent remedy. We complement that line of work by characterizing exactly how the most widely used post-hoc calibration map -- temperature scaling -- distorts the proxy. We prove an exact, model-free identity reducing the temperature-scaled proxy to the classifier's margin distribution, from which we obtain (i) strict monotonicity in the temperature and (ii) a continuous bijection from the temperature axis onto the open interval (0, 1/2), so that a fixed classifier -- with fixed decisions and fixed 0-1 error -- can be made to report any proxy value whatsoever. Under a Gaussian model of the logits we further derive a two-parameter closed form for the entire proxy-versus-temperature curve. Across CIFAR-10, Fashion-MNIST, and SVHN (eight binary tasks), the proxy varies by 56x to 980x at constant test error, the closed form reproduces the empirical curve to within 0.018, and the calibration temperature that minimizes the expected calibration error does not coincide with any stable proxy value. Our results give a precise, predictive account of the distortion whose existence motivates calibration-based remedies, and they reinforce the practical recommendation that a proxy value is meaningful only together with the mechanism that produced its probabilities.

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