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

Yang Yang, Chunlin Ji, Haoyang Li, Ke Deng

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.

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

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.

Paper Structure

This paper contains 27 sections, 24 theorems, 116 equations, 3 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Backgrounds and Preliminaries
  3. Single Gaussian Distribution
  4. Gaussian Mixture Model
  5. Energy Score
  6. Method
  7. NE-GMM with a Hybrid Loss
  8. Prediction with UQ based on NE-GMM
  9. Theoretical Results
  10. Experiments
  11. Toy Regressions
  12. Regression on UCI Datasets
  13. Financial time series forecasting
  14. Discussion
  15. Analysis of $S_e$ and its Derivatives
  16. ...and 12 more sections

Key Result

Proposition 1

The partial derivatives of $S_l(F_\psi(x),y)$ with respect to $\pi_k(x)$, $\mu_k(x)$, and $\sigma_k(x)$ in an IGMM are given by: where

Figures (3)

  • Figure 1: Predictive distribution plots for the toy regression example 1.
  • Figure 2: Predictive distribution plots for the toy regression example 2.
  • Figure 3: Trend plots of three stock datasets, where the red dashed lines in the figures indicate the time points dividing the training set and the test set.

Theorems & Definitions (29)

  • Proposition 1
  • Lemma 2
  • Theorem 3: Analytic form of energy score
  • Proposition 4: Partial derivatives of $S_e$
  • Lemma 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 9
  • Lemma 10
  • ...and 19 more