Prior-informed Uncertainty Modelling with Bayesian Polynomial Approximations

TL;DR

This paper introduces a Bayesian approach to polynomial chaos that incorporates prior knowledge and uncertainties, improving the accuracy of models used for uncertainty quantification in computational engineering.

Contribution

It presents a hierarchical Bayesian formulation for polynomial approximations that integrates expert knowledge and various priors to enhance predictive performance.

Findings

Bayesian polynomial approximations outperform traditional methods in predictive accuracy.

Incorporating expert priors improves model robustness and uncertainty quantification.

Hierarchical priors enable flexible modeling of physical and functional correlations.

Abstract

Orthogonal polynomial approximations form the foundation to a set of well-established methods for uncertainty quantification known as polynomial chaos. These approximations deliver models for emulating physical systems in a variety of computational engineering applications. In this paper, we describe a Bayesian formulation of polynomial approximations capable of incorporating uncertainties in input data. Through different priors in a hierarchical structure, this permits us to incorporate expert knowledge on the inference task via different approaches. These include beliefs of sparsity in the model; approximate knowledge of the polynomial coefficients (e.g. through low-fidelity estimates) or output mean, and correlated models that share similar functional and/or physical behaviours. We show that through a Bayesian framework, such prior knowledge can be leveraged to produce orthogonal polynomial approximations with enhanced predictive accuracy.