Within Loss functions
Why confident wrong answers hurt more
Cross-entropy loss penalises confident wrong classifications, helping models learn probabilities instead of just labels.
On this page
- Why classification loss measures confidence
- Spam filtering as a practical example
- How probability calibration changes learning
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Introduction
When a classification model learns, it is not enough to know whether an answer is right or wrong. It also matters how confident the model was. A prediction that assigns a 51% probability to the wrong class is a different kind of mistake from one that assigns 99.9% probability to the wrong class. Modern classification systems therefore use loss functions such as cross-entropy loss (also called log loss) that measure both correctness and confidence. These losses make highly confident mistakes far more expensive than uncertain ones, encouraging models to learn meaningful probabilities rather than simply choosing labels.[Scikit-learn+2ML Cheatsheet]scikit-learn.orgThis is the loss function used in (multinomial) logistic regression and extensions of it such as neural…Read more…
This idea is a key part of how loss functions turn mistakes into learning. By attaching larger penalties to overconfident errors, the training process pushes a model not only towards correct answers but also towards more trustworthy estimates of uncertainty.[Google for Developers+2Scikit-learn]google.comloss regularizationThe Log Loss equation returns the logarithm of the magnitude of the change, rather than just the …Read more
Why classification loss measures confidence
In many classification tasks, the model does not output a simple yes-or-no decision. Instead, it produces probabilities. For example, a spam filter might estimate:
- Spam: 90%
- Not spam: 10%
or
- Spam: 55%
- Not spam: 45%
Both predictions select “spam” as the final label, but they express very different levels of confidence.
Cross-entropy loss evaluates these probabilities directly. If the correct answer is spam and the model assigns a high probability to spam, the loss is small. If the model assigns a low probability to the correct class, the loss becomes larger. The penalty grows especially quickly when the model is extremely confident and wrong because the logarithmic form of the loss function increases sharply near probabilities of zero and one.[Scikit-learn+2Google for Developers]scikit-learn.orgThis is the loss function used in (multinomial) logistic regression and extensions of it such as neural…Read more…
Consider three predictions for an email that really is spam:
Predicted probability of spamOutcomeRelative loss0.90Correct and confidentLow0.60Correct but uncertainModerate0.01Wrong and extremely confidentVery high
The final case receives a dramatically larger penalty than the second. This design tells the model that being confidently wrong is worse than admitting uncertainty.[Medium+2Medium]medium.comCross-Entropy and Log Loss: Mathematical Foundations…Cross-Entropy strongly penalizes confident wrong predictions. That is, if y…
An important consequence is that two models can achieve the same classification accuracy while having very different losses. A model that gets most examples right but makes a few catastrophic, overconfident mistakes may have a worse loss score than a model that expresses more realistic uncertainty.[Cross Validated]stats.stackexchange.comDespite this, accuracy's value on validationCross ValidatedGood accuracy despite high loss value - Cross ValidatedJan 25, 2017 — During the training of a simple neural network binar…
Spam filtering as a practical example
Spam filtering illustrates why confidence-sensitive loss matters.
Imagine two systems evaluating the same message. The message is actually legitimate.
System A
- Spam probability: 51%
- Predicts spam
- Wrong, but only slightly confident
System B
- Spam probability: 99.9%
- Predicts spam
- Wrong and extremely confident[lightly.ai]lightly.aiA Brief Guide to Cross-Entropy LossWidely used in classification tasks, it penalizes confident wrong predictions and provides info…
If training treated both errors identically, the model would receive little information about the severity of the mistake. Cross-entropy instead gives System B a much larger penalty. The model learns that assigning near-certain probabilities should be reserved for situations where the evidence is genuinely overwhelming.[ML Cheatsheet+2Coralogix]ml-cheatsheet.readthedocs.ioML Cheatsheet Loss Functions — ML Glossary documentationML CheatsheetLoss Functions — ML Glossary documentation - Read the DocsCross-entropy loss, or log loss, measures the performance of a cla…
This behaviour is valuable in real systems because confidence often influences downstream decisions. An email service might automatically move messages with very high spam probabilities into a separate folder while leaving uncertain cases in the inbox. If the model’s confidence estimates are unreliable, users experience more frustrating errors. Confidence-aware loss functions help reduce that problem during training.[Scikit-learn]scikit-learn.org1.16. Probability calibrationThe calibration module allows you to better calibrate the probabilities of a given model, or to…
The same principle applies beyond spam detection:
- Medical image classifiers may use confidence scores to decide whether a human review is needed.
- Fraud-detection systems often prioritise investigations based on predicted probability.
- Content moderation systems may apply different actions depending on confidence levels.
In each case, a model that knows when it is uncertain is often more useful than one that merely produces the correct label slightly more often.[Scikit-learn]scikit-learn.org1.16. Probability calibrationThe calibration module allows you to better calibrate the probabilities of a given model, or to…
How probability calibration changes learning
A well-calibrated model has confidence scores that match reality. If it predicts 80% confidence across many examples, roughly 80% of those predictions should be correct. Calibration therefore concerns the quality of probabilities, not just classification accuracy.[Scikit-learn]scikit-learn.org1.16. Probability calibrationThe calibration module allows you to better calibrate the probabilities of a given model, or to…
Cross-entropy is widely used because it rewards probability estimates that align with observed outcomes. Researchers describe it as a “proper” loss, meaning that the best strategy is to predict probabilities that reflect true likelihoods rather than artificially exaggerated confidence.[arXiv]arxiv.orgImproving Calibration by Relating Focal Loss, Temperature Scaling, and PropernessAugust 21, 2024…
However, modern neural networks can still become overconfident despite being trained with cross-entropy. Researchers have shown that highly accurate models often produce confidence scores that are too high, especially on unfamiliar data. This has led to techniques such as temperature scaling and other calibration methods that adjust probabilities after training.[arXiv+2arXiv]arxiv.orgDon't Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary ClassificationFebruary 15…
The interaction between loss and calibration changes learning in several ways:[developers.google.com]developers.google.comregression: Loss | Machine Learning…
- Overconfident errors receive strong correction signals. The model is pushed to reduce certainty when certainty is not justified.[Lightly]lightly.aiA Brief Guide to Cross-Entropy LossWidely used in classification tasks, it penalizes confident wrong predictions and provides info…
- Moderately uncertain predictions receive smaller adjustments. Learning focuses attention where mistakes are most serious.[Medium]medium.comA Brief Overview of Cross Entropy Loss | by Chris HughesCross entropy loss is a mechanism to quantify how well a model's prediction…
- Probability estimates become useful outputs in their own right. The model learns not only what answer to choose but also how strongly to believe it.[LinkedIn]linkedin.comUnderstanding Log Loss For Classification EvaluationQuantifies Accuracy: It penalizes false classifications more heavily, making…
This distinction is one reason classification systems commonly optimise cross-entropy rather than a simple count of right and wrong answers. Accuracy only measures whether the final choice was correct. Cross-entropy measures how the model arrived at that choice and whether its confidence was justified.[Scikit-learn+2LinkedIn]scikit-learn.orgThis is the loss function used in (multinomial) logistic regression and extensions of it such as neural…Read more…
Why confident wrong answers hurt more
The central idea is straightforward: uncertainty is acceptable, but unjustified certainty is costly.
A model that says “I am 55% sure” and turns out to be wrong has expressed doubt. A model that says “I am 99.9% sure” and turns out to be wrong has made a much stronger claim. Cross-entropy loss reflects this difference mathematically by assigning far larger penalties to the second case.[Medium+2Medium]medium.comCross-Entropy and Log Loss: Mathematical Foundations…Cross-Entropy strongly penalizes confident wrong predictions. That is, if y…
Because training repeatedly minimises this loss, the model gradually learns to reserve extreme confidence for situations where the data truly supports it. The result is a classifier that not only predicts labels but also develops more informative probability estimates. In practical AI systems, those probability estimates are often as important as the final decision itself.[Lightly+2Scikit-learn]lightly.aiA Brief Guide to Cross-Entropy LossWidely used in classification tasks, it penalizes confident wrong predictions and provides info…
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Further Reading
Books and field guides related to Why confident wrong answers hurt more. Use these as the next step if you want deeper reading beyond the article.
Deep Learning
Rating: 3.5/5 from 6 Google Books ratings
Discusses log loss, probabilities, and training behavior.
Hands-on Machine Learning with Scikit-Learn, Keras, and Tenso...
Explains cross-entropy, confidence, and probabilistic predictions.
Pattern Recognition and Machine Learning
Strong treatment of probabilistic classification and uncertainty.
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Endnotes
1.
Source: scikit-learn.org
Link:https://scikit-learn.org/stable/modules/generated/sklearn.metrics.log_loss.html
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This is the loss function used in (multinomial) logistic regression and extensions of it such as neural...Read more...
2.
Source: lightly.ai
Link:https://www.lightly.ai/blog/cross-entropy-loss
Source snippet
A Brief Guide to Cross-Entropy LossWidely used in classification tasks, it penalizes confident wrong predictions and provides info...
3.
Source: developers.google.com
Title: loss regularization
Link:https://developers.google.com/[machine-learning
Source snippet
The Log Loss equation returns the logarithm of the magnitude of the change, rather than just the...Read more...
4.
Source: scikit-learn.org
Link:https://scikit-learn.org/stable/modules/calibration.html
Source snippet
1.16. Probability calibrationThe calibration module allows you to better calibrate the probabilities of a given model, or to...
5.
Source: coralogix.com
Link:https://coralogix.com/ai-blog/understanding-binary-cross-entropy-and-log-loss-for-effective-model-monitoring/
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6.
Source: medium.com
Link:https://medium.com/ai-enthusiast/cross-entropy-and-log-loss-mathematical-foundations-and-their-use-in-classification-eb708f9f629f
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Cross-Entropy and Log Loss: Mathematical Foundations...Cross-Entropy strongly penalizes confident wrong predictions. That is, if y...
7.
Source: koshurai.medium.com
Link:https://koshurai.medium.com/understanding-log-loss-the-math-behind-it-and-why-it-matters-for-machine-learning-success-22c10276560a
Source snippet
Log Loss measures how well a classification model predicts probabilities. It penalizes incorrect predictions more heavily when th...
8.
Source: scikit-learn.org
Link:https://scikit-learn.org/0.16/modules/calibration.html
Source snippet
1.16. Probability calibrationLogisticRegression returns well calibrated predictions by default as it directly optimizes log-loss. In cont...
9.
Source: arxiv.org
Title: arXiv Soft Calibration Objectives for Neural Networks
Link:https://arxiv.org/abs/2108.00106
10.
Source: arxiv.org
Link:https://arxiv.org/abs/2408.11598
Source snippet
Improving Calibration by Relating Focal Loss, Temperature Scaling, and PropernessAugust 21, 2024...
Published: August 21, 2024
11.
Source: arxiv.org
Link:https://arxiv.org/abs/2102.07856
Source snippet
Don't Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary ClassificationFebruary 15...
12.
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Link:https://medium.com/%40chris.p.hughes10/a-brief-overview-of-cross-entropy-loss-523aa56b75d5
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A Brief Overview of Cross Entropy Loss | by Chris HughesCross entropy loss is a mechanism to quantify how well a model's [prediction]({{ 'error-harms/' | relative_url }})...
13.
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Understanding Log Loss For Classification EvaluationQuantifies Accuracy: It penalizes false classifications more heavily, making...
14.
Source: scikit-learn.org
Title: model evaluation
Link:https://scikit-learn.org/stable/modules/model_evaluation.html
Source snippet
Metrics and scoring: quantifying the quality of predictionsThe sklearn.metrics module implements several loss, score, and utility functio...
15.
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16.
Source: developers.google.com
Title: logistic regression
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Regression | Machine Learning25 Aug 2025 — This course module teaches the fundamentals of logistic regression, including how to predict a...
17.
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Title: crash course
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18.
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19.
Source: scikit-learn.org
Title: model evaluation
Link:https://scikit-learn.org/1.0/modules/model_evaluation.html
Source snippet
Metrics and scoring: quantifying the quality of predictionsThe sklearn.metrics module implements several loss, score, and utility functio...
20.
Source: scikit-learn.org
Link:https://scikit-learn.org/stable/modules/linear_model.html
Source snippet
1.1. Linear ModelsFor multiclass classification, the problem is treated as multi-output regression, and the predicted class corresponds t...
21.
Source: scikit-learn.org
Link:https://scikit-learn.org/0.17/modules/generated/sklearn.metrics.log_loss.html
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sklearn.metrics.log_loss — scikit-learn 0.17.1 documentationLog loss is undefined for p=0 or p=1, so probabilities are clipped to max(eps...
22.
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That is, the loss is a number indicating how bad the model's prediction was on a single example. If the...Read more...
23.
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Title: Machine Learning crash course from Google(2)
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Published: May 30, 2019
24.
Source: medium.com
Title: Log Loss vs Cross Entropy
Link:https://medium.com/biased-algorithms/log-loss-vs-cross-entropy-740df12d7526
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25.
Source: koshurai.medium.com
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Link:https://koshurai.medium.com/understanding-log-loss-a-comprehensive-guide-with-code-examples-c79cf5411426
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26.
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Understanding Log Loss to measure model performance. If...Read more...
27.
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ML CheatsheetLoss Functions — ML Glossary documentation - Read the DocsCross-entropy loss, or log loss, measures the performance of a cla...
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Title: comscikit learn
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34.
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Additional References
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Source: github.com
Link:https://github.com/xbeat/Machine-Learning/blob/main/Explaining%20Log%20Loss%20Using%20Python.md
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Explaining Log Loss Using Python.mdLog loss, also known as logarithmic loss or cross-entropy loss, is a crucial metric in machine learnin...
36.
Source: merriam-webster.com
Link:https://www.merriam-webster.com/dictionary/log
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LOG Definition & Meaning5 days ago — The meaning of LOG is a usually bulky piece or length of a cut or fallen tree; especially: a length...
37.
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Link:https://www.youtube.com/watch?v=iBQlukGBZ78
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FREE Machine Learning Crash Course from Google... course: ✓ How does machine learning differ from traditional programming? ✓ What is loss...
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Machine Learning Crash Course: Logistic RegressionLogistic regression is a machine learning technique for predicting a probability. In th...
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Title: a practical guide to the loss function in machine learning
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40.
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Title: Common Loss Functions for Classification (Cross-Entropy)
Link:https://apxml.com/courses/introduction-to-[deep-learning
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0 \log(1-p) \to 0 log(1−p)→0, and the loss approaches 0. This formula effectively penalizes the model more heavily for confident wrong pr...
41.
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Title: binary cross entropy log loss for binary classification
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Binary Cross Entropy/Log Loss for Binary Classification24 Apr 2025 — Binary Cross Entropy is a loss function used in machine learning and...
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Title: cross entropy loss language models information theory
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Cross-Entropy Loss: Information Theory for Language...21 Feb 2026 — The loss decreases rapidly as model confidence increases, approachin...
43.
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rror for a single training example or a batch of examples...Read more...
44.
Source: github.com
Link:https://github.com/litaotao/machine-learning-crash-course
Source snippet
l's predictions was on a single example. Although MES is...Read more...
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