superior-scoring-rules 1.0.0

PBS and PLL are superior evaluation metrics for probabilistic classifiers, fixing flaws in Brier Score (MSE) and Log Loss (Cross-Entropy). Strictly proper, consistent, and better for model selection, early stopping, and checkpointing.

Superior Scoring Rules: Better Metrics for Probabilistic Evaluation (arXiv Preprint)

PBS and PLL are superior evaluation metrics for probabilistic classifiers, fixing flaws in Brier Score (MSE) and Log Loss (Cross-Entropy). Strictly proper, consistent, and better for model selection, early stopping, and checkpointing.

Problem with Traditional Metrics

Accuracy-based metrics (Accuracy, F1) treat all correct predictions equally, ignoring confidence. In high-stakes domains, confidence calibration is critical:

• Cancer Diagnosis: 51% vs. 99% confidence in malignancy should not be treated differently.

• ICU Triage & Mortality: Overconfident mispredictions risk patient safety.

• Autonomous Vehicles: Decisions depend on uncertainty about obstacles.

• Financial Risk Modeling: Pricing and investment hinge on calibrated probabilities.

• Security Threat Detection: High-confidence false negatives undermine defenses.

Thus, Accuracy or F1 Score alone is insufficient: they ignore the confidence of predictions.

Limitations of MSE & Cross-Entropy

Mean Squared Error (Brier Score) and Cross-Entropy (Log Loss) are strictly proper scoring rules, rewarding calibration. However, they can still favor incorrect predictions over correct ones. Example:

Vector	True Label (Y)	Predicted Probabilities (P)	Brier Score	Log Loss	State
A	[0, 1, 0]	[0.33, 0.34, 0.33]	0.6534	0.4685	Correct
B	[0, 1, 0]	[0.51, 0.49, 0.00]	0.5202	0.3098	Incorrect

Both MSE and Log Loss favor B over A, contradicting the principle of rewarding correct predictions.

Our Solution: PBS & PLL

To ensure correct predictions always receive better scores, we introduce a penalty term for misclassifications:

• Penalized Brier Score (PBS)

• Penalized Logarithmic Loss (PLL)

These metrics are both strictly proper and superior (never favor wrong over right).

Quick Start

Installation from PyPI

pip install superior-scoring-rules

Install from Source (Development)

Clone the repository:

git clone https://github.com/Ruhallah93/superior-scoring-rules.git

Basic Usage

import tensorflow as tf
from superior_scoring_rules import pbs, pll

# Sample data (batch_size=3, num_classes=4)
y_true = tf.constant([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]])
y_pred = tf.constant([[0.9, 0.05, 0.05, 0], 
                     [0.1, 0.8, 0.05, 0.05],
                     [0.1, 0.1, 0.1, 0.7]])

print("PBS:", pbs(y_true, y_pred).numpy())
print("PLL:", pll(y_true, y_pred).numpy())

Early Stopping & Checkpointing

Use PBS/PLL instead of val_loss:

class PBSCallback(tf.keras.callbacks.Callback):
    def on_epoch_end(self, epoch, logs=None):
        logs = logs or {}
        logs['val_pbs'] = pbs(self.validation_data[1], self.model.predict(self.validation_data[0]))
        # or
        logs['val_pll'] = pll(self.validation_data[1], self.model.predict(self.validation_data[0]))

model.fit(..., callbacks=[PBSCallback(),
    tf.keras.callbacks.EarlyStopping(monitor='val_pbs', patience=5, mode='min'),
    tf.keras.callbacks.ModelCheckpoint('best.h5', monitor='val_pbs', save_best_only=True)
])

Paper & Citation

• Superior scoring rules for probabilistic evaluation of single-label multi-class classification tasks

• arXiv: 2407.17697

@article{ahmadian2025superior,
  title={Superior scoring rules for probabilistic evaluation of single-label multi-class classification tasks},
  author={Ahmadian, Rouhollah and Ghatee, Mehdi and Wahlstr{\"o}m, Johan},
  journal={International Journal of Approximate Reasoning},
  pages={109421},
  year={2025},
  publisher={Elsevier}
}