Energy Score-Guided Neural Gaussian Mixture Model for Predictive Uncertainty Quantification

2026-03-29

AI Summary

This work addresses the bias in mean and variance estimation commonly observed in existing parametric approaches to uncertainty quantification, which often suffer from training instability and mode collapse. The authors propose the Neural Energy Gaussian Mixture Model (NE-GMM), which uniquely integrates energy scores (ES) with Gaussian mixture models (GMMs) to construct a strictly proper scoring rule framework, accompanied by a theoretically derived generalization error bound. NE-GMM employs neural networks to parameterize the GMM and optimizes a hybrid loss combining energy scores and negative log-likelihood, enabling effective modeling of complex multimodal distributions and improved calibration. Experimental results demonstrate that NE-GMM consistently outperforms current baselines on both synthetic and real-world datasets, achieving superior performance in predictive accuracy and uncertainty calibration.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Quantifying predictive uncertainty is essential for real world machine learning applications, especially in scenarios requiring reliable and interpretable predictions. Many common parametric approaches rely on neural networks to estimate distribution parameters by optimizing the negative log likelihood. However, these methods often encounter challenges like training instability and mode collapse, leading to poor estimates of the mean and variance of the target output distribution. In this work, we propose the Neural Energy Gaussian Mixture Model (NE-GMM), a novel framework that integrates Gaussian Mixture Model (GMM) with Energy Score (ES) to enhance predictive uncertainty quantification. NE-GMM leverages the flexibility of GMM to capture complex multimodal distributions and leverages the robustness of ES to ensure well calibrated predictions in diverse scenarios. We theoretically prove that the hybrid loss function satisfies the properties of a strictly proper scoring rule, ensuring alignment with the true data distribution, and establish generalization error bounds, demonstrating that the model's empirical performance closely aligns with its expected performance on unseen data. Extensive experiments on both synthetic and real world datasets demonstrate the superiority of NE-GMM in terms of both predictive accuracy and uncertainty quantification.

Problem

• predictive uncertainty quantification
• training instability
• mode collapse
• Gaussian Mixture Model
• Energy Score

Innovation

• Neural Energy Gaussian Mixture Model
• Energy Score
• Predictive Uncertainty Quantification
• Strictly Proper Scoring Rule
• Generalization Error Bound