# Superior Scoring Rules for Probabilistic Evaluation of Single-Label Multi-Class Classification Tasks

arXiv: https://arxiv.org/abs/2407.17697 · HTML: https://mlsanity.com/paper/2407.17697/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 46 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 7 of 7)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Deep Learning for Spatio-Temporal Data Mining: A Survey](https://mlsanity.com/paper/1906.04928/) (2019)
- [Evaluating model calibration in classification](https://mlsanity.com/paper/1902.06977/) (2019)
- [A Case Study Competition Among Methods for Analyzing Large Spatial Data](https://mlsanity.com/paper/1710.05013/) (2017)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Local scale invariance and robustness of proper scoring rules](https://mlsanity.com/paper/1912.05642/) (2019)
- [From Classification Accuracy to Proper Scoring Rules: Elicitability of Probabilistic Top List Predictions](https://mlsanity.com/paper/2301.11797/) (2023)

## Built on by (most cited first, 1 of 1)

- [Development and Comparative Evaluation of Three Artificial Intelligence Models (NLP, LLM, JEPA) for Predicting Triage in Emergency Departments: A 7-Month Retrospective Proof-of-Concept](https://mlsanity.com/paper/2507.01080/) (2025)

## Abstract (arXiv)

This study introduces novel superior scoring rules called Penalized Brier Score (PBS) and Penalized Logarithmic Loss (PLL) to improve model evaluation for probabilistic classification. Traditional scoring rules like Brier Score and Logarithmic Loss sometimes assign better scores to misclassifications in comparison with correct classifications. This discrepancy from the actual preference for rewarding correct classifications can lead to suboptimal model selection. By integrating penalties for misclassifications, PBS and PLL modify traditional proper scoring rules to consistently assign better scores to correct predictions. Formal proofs demonstrate that PBS and PLL satisfy strictly proper scoring rule properties while also preferentially rewarding accurate classifications. Experiments showcase the benefits of using PBS and PLL for model selection, model checkpointing, and early stopping. PBS exhibits a higher negative correlation with the F1 score compared to the Brier Score during training. Thus, PBS more effectively identifies optimal checkpoints and early stopping points, leading to improved F1 scores. Comparative analysis verifies models selected by PBS and PLL achieve superior F1 scores. Therefore, PBS and PLL address the gap between uncertainty quantification and accuracy maximization by encapsulating both proper scoring principles and explicit preference for true classifications. The proposed metrics can enhance model evaluation and selection for reliable probabilistic classification.