---
kind: concept
slug: temperature-scaling
title: Temperature scaling
summary: >-
  Divides a softmax model's logits by one learned positive number after training, without changing the top class. It is the one-parameter case of Dirichlet calibration's linear family. On average ranks it trails Dirichlet ODIR on deep networks and Dirichlet L2 on non-neural classifiers. LogitNorm's training-time temperature is a different use. Sources are author-reported.
tags: [calibration, uncertainty, classification]
papers: ["1910.12656", "2205.09310"]
related: [calibration, uncertainty-estimation, ensembles, out-of-distribution-detection]
status: draft
updated_at: 2026-10-31
---

## Overview

A classifier that outputs logits can be over-confident. Temperature scaling
fixes this after training by dividing every logit by one number t > 0 before
the softmax. The top class stays the same, because dividing by a positive number
keeps the order of the logits, but the confidence changes. The calibration
fixes for neural networks (temperature, matrix and vector scaling) act on logits
and assume a softmax output layer [arxiv:1910.12656 §1].

The Dirichlet calibration paper places temperature scaling inside a wider
family. Temperature scaling is the case W = I/t, b = 0, applied to the
probability vector softmax(z) [arxiv:1910.12656 §3] [app. B, Prop. 1]. The
paper's other members of the family are described in the variants below.

Temperature also appears inside a training loss. LogitNorm uses a temperature
during training rather than fitting one afterwards, so it is a different
method that shares the word. It is described in its own variant below.

## Variants

### Temperature scaling (one scalar on logits)

One parameter t, fitted on held-out data. The paper calls this a special case
of its family, not a separate method [arxiv:1910.12656 §3].
Results are author-reported. On 21 UCI datasets with 11 classifiers, the
average log-loss rank is 4.61, against 2.25 for Dirichlet L2 and 2.92 for beta
calibration; uncalibrated models rank 5.54 [arxiv:1910.12656 tab. 2]. On the
p-cw-ECE measure, temperature scaling ranks 4.37, against 2.34 for Dirichlet L2
[arxiv:1910.12656 tab. 1]. On 14 deep-network cases, it has the average cw-ECE
rank 3.71 among the three general-purpose methods, against 3.79 for Dirichlet
L2 and 2.79 for Dirichlet ODIR [arxiv:1910.12656 tab. 3].

### Vector and matrix scaling (per-class parameters on logits)

The card names vector and matrix scaling as logit-based fixes [arxiv:1910.12656
§1] and says they are not general-purpose [arxiv:1910.12656 §4.2]. In the
deep-network study, vector scaling has the best average cw-ECE rank of all six
methods, 2.29 [arxiv:1910.12656 tab. 3]. Matrix scaling with ODIR regularisation
(MS-ODIR) has the best average log-loss rank, 1.64 [arxiv:1910.12656 tab. 4].
The card does not define these methods. Their definitions are in Guo et al.
(ICML 2017), which has no card here, so this page does not describe them
further.

### Dirichlet calibration (linear map on log-probabilities)

Card: [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](/papers/1910.12656).
The method takes any probability vector, not only logits, and applies a
k × k matrix and a bias to its logarithm, followed by a softmax [arxiv:1910.12656
§3]. It has k² + k parameters, so 10,100 for 100 classes [arxiv:1910.12656 §3].
Regularisation comes in two forms: L2 (Dirichlet L2) and off-diagonal and
intercept regularisation (ODIR). On 21 UCI datasets, Dirichlet L2 has the best
average log-loss rank, 2.25 [arxiv:1910.12656 tab. 2]. On deep networks, Dirichlet
ODIR has the best average cw-ECE rank of the three general-purpose methods, 2.79,
and an average log-loss rank of 2.93, against 3.50 for temperature scaling
[arxiv:1910.12656 tab. 3] [arxiv:1910.12656 tab. 4]. The paper says this
difference in the deep-network results is "without statistical significance"
[arxiv:1910.12656 §4.2].

### Training-time temperature (LogitNorm)

Not a post-hoc fit. LogitNorm replaces cross-entropy on raw logits with
cross-entropy on logits divided by their L2 norm and by a temperature tau. Each
training logit vector then has length 1/tau [arxiv:2205.09310 §3.2] [eq. 4].
Tau is a training hyperparameter, tuned on Gaussian noise used as an OOD
validation set [arxiv:2205.09310 §4.1]. The main evaluation is OOD detection.
The softmax score's average FPR95 on CIFAR-10 with WRN-40-2 falls from 49.52%
to 15.65% over six OOD sets [arxiv:2205.09310 tab. 1]. Calibration is reported
in one table. Without temperature scaling, the LogitNorm model's ECE on CIFAR-10
is 66.95%, against 3.20% for cross-entropy. With temperature scaling, it is
0.41%, against 0.66% [arxiv:2205.09310 tab. 4]. These are author-reported, on
one architecture, and the text does not discuss the uncalibrated row
[arxiv:2205.09310 §4.2]. Card: [Mitigating Neural Network Overconfidence with
Logit Normalization](/papers/2205.09310).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Softmax model, logits available, and you want the simplest fix with one parameter | Temperature scaling | One parameter, and a special case of the family [arxiv:1910.12656 §3]. Trails Dirichlet ODIR on average ranks for deep networks [arxiv:1910.12656 tab. 3] [arxiv:1910.12656 tab. 4]. |
| Only probability outputs, or a non-softmax classifier | Dirichlet L2 | The method takes any probability vector [arxiv:1910.12656 §1]. On non-neural classifiers, Dirichlet L2 has the best average p-cw-ECE rank, 2.34, against 4.37 for temperature scaling [arxiv:1910.12656 tab. 1]. |
| Deep network, general-purpose calibrator, and you can fit k² + k parameters | Dirichlet ODIR | Best average cw-ECE rank of the three general-purpose methods on deep networks [arxiv:1910.12656 tab. 3]. The paper gives no minimum held-out size. |
| Logits available and you accept a method the paper does not count as general-purpose | Matrix scaling with ODIR | Best average log-loss rank of all six methods on deep networks [arxiv:1910.12656 tab. 4]. |
| Limited implementation time or tuning budget | Temperature scaling | The Dirichlet fit has several implementation choices that change results; see the card's Paper vs code and Pitfalls sections [arxiv:1910.12656 §3]. |
| Model already trained, and you cannot retrain it | Temperature scaling | Fitted after training on held-out data, so no retraining is needed [arxiv:1910.12656 §3]. Dirichlet maps are fitted the same way, with more parameters. |
| You control training, want less OOD over-confidence, and can afford several training runs | LogitNorm, with temperature scaling applied afterwards | Author-reported FPR95 gains on CIFAR-10 and CIFAR-100 [arxiv:2205.09310 tab. 1]. Its ECE is reported with temperature scaling applied [arxiv:2205.09310 tab. 4]. Tuning tau is expensive [arxiv:2205.09310 app. Future Work]. |

Fit any of these on held-out data, not training data. The Dirichlet loss is
defined over validation instances [arxiv:1910.12656 §3]. Fitting on training data
would not measure the miscalibration the method corrects.

## Implementation sketch

The card gives the log-loss fit for the whole family [arxiv:1910.12656 §3].
Applying it with W = I/t and b = 0 is this page's reading, not a separate result
in the card. Dividing log-probabilities by t gives the same softmax as dividing
logits by t, because the softmax removes the constant log-normaliser.

```pseudocode
# Temperature scaling as the one-parameter case of the linear form [§3]
# Input:  logits z (n, k) from a softmax model; labels y (n,) on held-out data
# Output: scalar t > 0; calibrated probabilities are softmax(z / t)

def fit_temperature(z, y):
    # Minimise log-loss over t > 0, on validation data only  [§3, "Fitting"]
    # Parametrise by log t so that t stays positive
    ...
    return t

def calibrate(z, t):
    # Same as W = I / t, b = 0 in softmax(W ln q + b) with q = softmax(z)  [§3]
    return softmax(z / t)
```

## Contradictions

The paper's text and its average ranks point in different directions on
Dirichlet ODIR against temperature scaling. The text says Dirichlet ODIR beats
temperature scaling on cw-ECE "at a small log-loss cost" [arxiv:1910.12656
§4.2]. The average log-loss ranks put Dirichlet ODIR ahead, 2.93 against 3.50
[arxiv:1910.12656 tab. 4]. Both are the paper's results. This page does not
resolve them.

The two papers' calibration numbers are not comparable. The Dirichlet study
reports average ranks over 21 UCI datasets and 14 deep-network cases
[arxiv:1910.12656 tab. 3]. The LogitNorm table gives one ECE per setting, for
CIFAR-10 and CIFAR-100 on WRN-40-2 [arxiv:2205.09310 tab. 4]. This page does
not combine them.

## Papers

- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](/papers/1910.12656): places temperature scaling as a special case of a wider family, with deep-network and non-neural comparisons.
- [Mitigating Neural Network Overconfidence with Logit Normalization](/papers/2205.09310): uses a temperature inside the training loss for OOD detection; reports ECE with and without post-hoc temperature scaling.
- Guo et al., "On Calibration of Modern Neural Networks", ICML 2017: the source of temperature, vector and matrix scaling as named in the card. No card here, so nothing from it is used.