A Gaussian process and linear-based framework for computing cut distributions in modular Bayesian calibration of two chained computer models

Computer models are widely used in science and engineering to simulate complex systems. However, these models are affected by several sources of uncertainty, which may limit their use for decision making in risk management. We present a Bayesian approach for quantifying parameter uncertainty in a chain of two computer models motivated by multiphysics simulations in the nuclear field. Part of the inputs of a downstream model parametrized by $\theta \in \mathbb{R}^p$ come from the outputs of an upstream model parametrized by $\lambda \in \mathbb{R}^q$. Usually, the joint posterior distribution of $(\theta, \lambda)$ would be obtained by applying Bayes' theorem using the experimental observations of both models. However, when the observations of the downstream model are too indirect to provide informative inference on $\lambda$, it may be preferable to compute a modular posterior distribution of $(\theta, \lambda)$, referred to as the \emph{cut distribution}. Assuming that the posterior distribution of $\lambda$ has been previously estimated from observations of the upstream model only, we aim to compute the posterior distribution of $\theta$ conditional on $\lambda$ using observations from the downstream model. To this end, we propose a Gaussian-process and linear-based framework to estimate the functional dependence between $\theta$ and $\lambda$, denoted by $\theta(\lambda)$, where each component is modeled as a realization of a Gaussian process. As the downstream model is approximated by a linear function of $\theta(\lambda)$, Bayesian conjugacy allows us to derive a Gaussian posterior predictive distribution of $\theta(\lambda)$ for any realization of $\lambda$. The effectiveness of the method is illustrated through several synthetic examples, and we highlight how variations in $\lambda$ impact the predictive distribution of the chained simulation.