Bayesian Model Calibration for Extrapolative Prediction via Gibbs Posteriors

TL;DR

This paper introduces Gibbs posteriors for model calibration, focusing on extrapolative prediction of physical parameters, addressing issues of traditional Gaussian process methods, and providing a modular, robust approach with practical implementation.

Contribution

It proposes Gibbs posteriors as an alternative to Gaussian process calibration, enabling uncertainty quantification without modeling discrepancy, with methods for tuning and combining posteriors.

Findings

Gibbs posteriors achieve nominal frequentist coverage.

The method is highly modular and adaptable.

Application to tantalum data demonstrates effectiveness.

Abstract

The current standard Bayesian approach to model calibration, which assigns a Gaussian process prior to the discrepancy term, often suffers from issues of unidentifiability and computational complexity and instability. When the goal is to quantify uncertainty in physical parameters for extrapolative prediction, then there is no need to perform inference on the discrepancy term. With this in mind, we introduce Gibbs posteriors as an alternative Bayesian method for model calibration, which updates the prior with a loss function connecting the data to the parameter. The target of inference is the physical parameter value which minimizes the expected loss. We propose to tune the loss scale of the Gibbs posterior to maintain nominal frequentist coverage under assumptions of the form of model discrepancy, and present a bootstrap implementation for approximating coverage rates. Our approach is highly modular, allowing an analyst to easily encode a wide variety of such assumptions. Furthermore, we provide a principled method of combining posteriors calculated from data subsets. We apply our methods to data from an experiment measuring the material properties of tantalum.