Embedded model form uncertainty quantification with measurement noise for Bayesian model calibration

Abstract A key factor in ensuring the accuracy of computer simulations that model physical systems is the proper calibration of their parameters based on real-world observations or experimental data. Bayesian methods provide a robust framework for quantifying and propagating the uncertainties that inevitably arise. Nevertheless, they produce predictions unable to represent the observed datapoints when paired with inexact models. Additionally, the quantified uncertainties of these overconfident models cannot be propagated to other Quantities of Interest (QoIs) reliably. A promising solution involves embedding a model inadequacy term in the inference parameters, allowing the quantified model form uncertainty to influence non-observed QoIs. In this work, we revisit this embedded formulation and analyze how different likelihood constructions affect the inference of model form uncertainty, particularly under the presence of prescribed measurement noise and unavoidable model discrepancies. Two additional likelihood formulations, the global moment-matching and relative global moment-matching likelihoods, are introduced to explore alternative ways of representing the residual distribution. The behavior of these likelihoods is examined alongside existing formulations to show how different treatments of measurement noise and discrepancies shape the inferred parameter posteriors, and thereby affect the uncertainty ultimately propagated to the QoIs. Particular attention is given to how the uncertainty associated with the model inadequacy term propagates to the QoIs for the posteriors obtained from different likelihood formulations, enabling a more comprehensive statistical analysis of the prediction’s reliability. Finally, the proposed approach is applied to estimate the uncertainty in the predicted heat flux from a transient thermal simulation using temperature observations.