The Well-Tempered Classifier: Some Elementary Properties of Temperature Scaling

Pierre-Alexandre Mattei; Bruno Loureiro

stat.MLFeb 16, 2026

The Well-Tempered Classifier: Some Elementary Properties of Temperature Scaling

Authors: Pierre-Alexandre Mattei, Bruno Loureiro

Organizations: Université Côte d’Azur, Inria, CNRS, LJAD, France · Département d’Informatique, École Normale Supérieure - PSL, CNRS, France

Abstract

Temperature scaling is a simple method that allows to control the uncertainty of probabilistic models. It is mostly used in two contexts: improving the calibration of classifiers and tuning the stochasticity of large language models (LLMs). In both cases, temperature scaling is the most popular method for the job. Despite its popularity, a rigorous theoretical analysis of the properties of temperature scaling has remained elusive. We investigate here some of these properties. For classification, we show that increasing the temperature increases the uncertainty in the model in a very general sense (and in particular increases its entropy). However, for LLMs, we challenge the common claim that increasing temperature increases diversity. Furthermore, we introduce two new characterisations of temperature scaling. The first one is geometric: the tempered model is shown to be the information projection of the original model onto the set of models with a given entropy. The second characterisation clarifies the role of temperature scaling as a submodel of more general linear scalers such as matrix scaling and Dirichlet calibration: we show that temperature scaling is the only linear scaler that does not change the hard predictions of the model.

Figures & tables

Figure 1: Entropies of two tempered versions of a toy language model. Standard temperature scaling has the expected monotonic effect on entropy, but is impractical for LLMs. The “myopic" version, widely used in practice, has a much less intuitive behaviour.

Figure 2: Temperature scaling as an information projection. In both cases, the initial model

p

is projected onto the set

\mathcal{Q}

of distributions of constant entropy

h^{\star}=0.9

. The blue arrow is the path of all tempered models between

p

and its projection

p_{\beta^{\star}}

. This path can be interpreted as the geodesic between

p

and

p_{\beta^{\star}}

. (Left) The initial distribution

p=(0.01,0.09,0.9)

has a lower entropy than

h^{\star}

, and is warmed up towards

p_{\beta^{\star}}

with

\beta^{\star}\approx 0.37

. (Right) The initial distribution

p=(0.4,0.35,0.25)

has a higher entropy than

h^{\star}

, and is cooled down towards

p_{\beta^{\star}}

with

\beta^{\star}\approx 3.98

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Terms involved in the computation of the entropy of the myopically scaled model. (Left) The marginal and conditional entropies of Equation 44 . (Right) The two terms of the decomposition of Equation ( 45 ). The source of nonmotonicity is the term

\pi_{\beta}H(p_{\beta}(x_{2}|x_{1}=1)

, that is the product of an increasing and a decreasing functions.

Model Calibration

Model calibration aligns predicted confidence or model outputs with observed outcomes and measurements to support more reliable decisions.

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Improving Semantic Uncertainty Quantification in Language Model Question-Answering via Token-Level Temperature Scaling

Tom A. Lamb, Desi R. Ivanova, Philip H. S. Torr, Tim G. J. Rudner

University of Oxford · University of Toronto & Vijil

Language Model Calibration LLM Uncertainty Estimation Semantic Uncertainty Estimation Model Calibration Question Answering

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Calibration is central to reliable semantic uncertainty quantification, yet prior work has largely focused on discrimination, neglecting calibration. As calibration and discrimination capture distinct aspects of uncertainty, focusing on discrimination alone yields an incomplete picture. We address this gap by systematically evaluating both aspects across a broad set of confidence measures. We show that current approaches, particularly fixed-temperature heuristics, produce systematically miscalibrated and poorly discriminative semantic confidence distributions. We demonstrate that optimising a single scalar temperature, which, we argue, provides a suitable inductive bias, is a surprisingly simple yet effective solution. Our exhaustive evaluation confirms that temperature scaling consistently improves semantic calibration, discrimination, and downstream entropy, outperforming both heuristic baselines and more expressive token-level recalibration methods on question-answering tasks.

Abstract

Model Calibration

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