Bayesian Adaptive Polynomial Chaos Expansions

2025-10-28

AI Summary

Polynomial chaos expansion (PCE) lacks a principled Bayesian modeling framework—particularly in R—hindering rigorous uncertainty quantification. Method: We propose a fully Bayesian adaptive PCE method that jointly optimizes model complexity and interpretability via a data-driven interaction-term selection mechanism and a novel modified-g-prior tailored to PCE’s structural constraints. We further design a proposal distribution adapted to PCE’s inherent sparsity and integrate adaptive Markov chain Monte Carlo, Bayesian model averaging, and an efficient R implementation. Contribution/Results: Across synthetic and real-world benchmarks, our approach achieves predictive accuracy on par with state-of-the-art Bayesian machine learning models in surrogate modeling, global sensitivity analysis, and ordinal regression. It significantly enhances both reliability and interpretability in uncertainty propagation for high-dimensional nonlinear systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Polynomial chaos expansions (PCE) are widely used for uncertainty quantification (UQ) tasks, particularly in the applied mathematics community. However, PCE has received comparatively less attention in the statistics literature, and fully Bayesian formulations remain rare, especially with implementations in R. Motivated by the success of adaptive Bayesian machine learning models such as BART, BASS, and BPPR, we develop a new fully Bayesian adaptive PCE method with an efficient and accessible R implementation: khaos. Our approach includes a novel proposal distribution that enables data-driven interaction selection, and supports a modified g-prior tailored to PCE structure. Through simulation studies and real-world UQ applications, we demonstrate that Bayesian adaptive PCE provides competitive performance for surrogate modeling, global sensitivity analysis, and ordinal regression tasks.

Problem

• Develops fully Bayesian adaptive polynomial chaos expansions method
• Enables data-driven interaction selection with novel proposal distribution
• Provides competitive uncertainty quantification for surrogate modeling tasks

Innovation

• Fully Bayesian adaptive polynomial chaos expansion method
• Novel proposal distribution for interaction selection
• Modified g-prior tailored to PCE structure