Hierarchical Bayesian Calibration with Bayesian Committee Machine

Abstract

Calibrating computational models to experimental data is a core task in applied statistics, especially in scientific domains, where physical experiments are costly and simulations play a central role in design and inference. Motivated by uncertainty quantification challenges in particle accelerator experiments, we develop and evaluate a Hierarchical Bayesian Calibration framework. In contrast to standard Bayesian calibration, certain inputs - such as beam injection amplitude - must be estimated separately for each experiment. We adopt the Kennedy-O'Hagan formulation and extend it with a hierarchical prior structure to model the distribution of experiment-specific calibration parameters, thus borrowing strength and improving generalisation across repeated experiments. A key methodological challenge arises from the need to evaluate a large number of forward simulations, which renders conventional Markov chain Monte Carlo approaches computationally prohibitive. To address this, we leverage the Bayesian Committee Machine as a scalable modelling strategy for Gaussian Process emulators. The BCM provides a principled divide-and-conquer approach, enabling parallel inference and reducing computational cost without requiring problem-specific tuning of the emulator approximation. Posterior sampling is performed using the No-U-Turn Sampler, supported by automatic differentiation in Julia, which removes the need for analytic gradient derivation and facilitates flexible model specification. We assess the proposed framework using established benchmark problems and simulated data from the Argonne Wakefield Accelerator. The results demonstrate substantial computational savings and robust calibration performance, highlighting the applicability of the method to large-scale scientific modelling problems.