# On Focal Loss for Class-Posterior Probability Estimation: A Theoretical Perspective

arXiv: https://arxiv.org/abs/2011.09172 · HTML: https://mlsanity.com/paper/2011.09172/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Nov 2020.

## Built on by (most cited first, 21 of 21)

- [Dual Focal Loss for Calibration](https://mlsanity.com/paper/2305.13665/) (2023)
- [A Short Survey on Importance Weighting for Machine Learning](https://mlsanity.com/paper/2403.10175/) (2024)
- [Realistic Evaluation of Deep Partial-Label Learning Algorithms](https://mlsanity.com/paper/2502.10184/) (2025)
- [Being Properly Improper](https://mlsanity.com/paper/2106.09920/) (2021)
- [Revisiting Confidence Estimation: Towards Reliable Failure Prediction](https://mlsanity.com/paper/2403.02886/) (2024)
- [On Calibration of Graph Neural Networks for Node Classification](https://mlsanity.com/paper/2206.01570/) (2022)
- [RankSEG: A Consistent Ranking-based Framework for Segmentation](https://mlsanity.com/paper/2206.13086/) (2022)
- [Neural Clamping: Joint Input Perturbation and Temperature Scaling for Neural Network Calibration](https://mlsanity.com/paper/2209.11604/) (2022)
- [Improving Calibration by Relating Focal Loss, Temperature Scaling, and Properness](https://mlsanity.com/paper/2408.11598/) (2024)
- [Diversity in Large Language Models under Supervised Fine-Tuning](https://mlsanity.com/paper/2605.00195/) (2026)
- [Uncertainty Calibration for Deep Audio Classifiers](https://mlsanity.com/paper/2206.13071/) (2022)
- [Beyond One-Hot Labels: Semantic Mixing for Model Calibration](https://mlsanity.com/paper/2504.13548/) (2025)
- [h-calibration: Rethinking Classifier Recalibration with Probabilistic Error-Bounded Objective](https://mlsanity.com/paper/2506.17968/) (2025)
- [AugLoss: A Robust Augmentation-based Fine Tuning Methodology](https://mlsanity.com/paper/2206.02286/) (2022)
- [PostHoc FREE Calibrating on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2503.01195/) (2025)
- [Revisiting Reweighted Risk for Calibration: AURC, Focal, and Inverse Focal Loss](https://mlsanity.com/paper/2505.23463/) (2025)
- [Functional Properties of the Focal-Entropy](https://mlsanity.com/paper/2603.02533/) (2026)
- [Focused PU learning from imbalanced data](https://mlsanity.com/paper/2605.14467/) (2026)
- [Structured Proper Loss Geometries for Multiclass Classification: Theory and Controlled Empirical Evaluation](https://mlsanity.com/paper/2606.29471/) (2026)
- [Audit Before You Update: Recording Artifacts and the Limits of Bounded Model Updates in MIMIC-IV ICU Prediction](https://mlsanity.com/paper/2607.19020/) (2026)
- [Recent advances in weakly supervised learning: New supervision paradigms, assumption relaxations, and practical solutions](https://mlsanity.com/paper/2608.06896/) (2026)

## Abstract (arXiv)

The focal loss has demonstrated its effectiveness in many real-world applications such as object detection and image classification, but its theoretical understanding has been limited so far. In this paper, we first prove that the focal loss is classification-calibrated, i.e., its minimizer surely yields the Bayes-optimal classifier and thus the use of the focal loss in classification can be theoretically justified. However, we also prove a negative fact that the focal loss is not strictly proper, i.e., the confidence score of the classifier obtained by focal loss minimization does not match the true class-posterior probability and thus it is not reliable as a class-posterior probability estimator. To mitigate this problem, we next prove that a particular closed-form transformation of the confidence score allows us to recover the true class-posterior probability. Through experiments on benchmark datasets, we demonstrate that our proposed transformation significantly improves the accuracy of class-posterior probability estimation.