A Framework for the Bayesian Calibration of Complex and Data-Scarce Models in Applied Sciences

TL;DR

This work surveys Bayesian calibration theory for complex, data-scarce computer models and introduces a unified Kennedy–O’Hagan framework augmented with Gaussian-process surrogates and multi-output capabilities. It presents ACBICI, an open-source Python library that implements single- and multi-output calibration with diagnostics, sensitivity analysis, and surrogate-based inference (MCMC and VBMC). The authors provide practical guidelines on scaling, priors, convergence diagnostics, and interpretability, bridging theory and real-world application. Through gravity, Cobb–Douglas, and traction-test examples, the paper demonstrates robust uncertainty quantification, predictive performance, and the utility of discrepancy modeling when models are imperfect. Overall, the framework and software aim to make rigorous Bayesian calibration accessible for data-scarce, computationally intensive engineering problems across disciplines.

Abstract

In this work, we review the theory involved in the Bayesian calibration of complex computer models, with particular emphasis on their use for applications involving computationally expensive simulations and scarce experimental data. In the article, we present a unified framework that incorporates various Bayesian calibration methods, including well-established approaches. Furthermore, we describe their implementation and use with a new, open-source Python library, ACBICI (A Configurable BayesIan Calibration and Inference Package). All algorithms are implemented with an object-oriented structure designed to be both easy to use and readily extensible. In particular, single-output and multiple-output calibration are addressed in a consistent manner. The article completes the theory and its implementation with practical recommendations for calibrating the problems of interest. These guidelines -- currently unavailable in a unified form elsewhere -- together with the open-source Python library, are intended to support the reliable calibration of computational codes and models commonly used in engineering and related fields. Overall, this work aims to serve both as a comprehensive review of the statistical foundations and (computational) tools required to perform such calculations, and as a practical guide to Bayesian calibration with modern software tools.

Figures (23)

• Figure 1: Structure of a Bayesian calibration procedure.
• Figure 2: Example \ref{['ss-a-mcmc']}. Comparison between experimental data (blue markers) and calibrated model predictions (red line). Close agreement indicates accurate reproduction of the observed behavior using the MAP parameter estimates.
• Figure 3: Example \ref{['ss-a-mcmc']}. Corner plot showing marginal and joint posterior distributions of the calibrated parameters. Diagonal panels display marginal densities with MAP, mean, median, and 95% credible intervals; off-diagonal panels illustrate parameter correlations.
• Figure 4: Example \ref{['ss-c-bayes']}. Comparison between experimental data (markers) and model predictions after discrepancy calibration. The discrepancy-corrected predictions provide improved agreement with the observations, particularly in regions where the base model exhibits systematic deviations.
• Figure 5: Example \ref{['ss-c-bayes']}. Estimated model discrepancy using a Gaussian process surrogate, reflecting systematic deviations between the model and data.