Options for the inclusion of model discrepancy in Bayesian calibration

One of the challenging issues in the well-known Kennedy and O’Hagan framework for Bayesian calibration is to formulate the prior of model discrepancy function, which can significantly affect the results of calibration. In the absence of physical knowledge on model inadequacy, it is often not clear how to construct a suitable prior, whereas an inappropriate selection of prior may lead to biased or useless parameter estimation. Aiming to address the uncertainty arising from the selection of a particular prior, this paper conducts an extensive study on possible formulations of model discrepancy function, and proposes a three-step (calibration, validation, and combination) approach in order to inform the decision on the construction of model discrepancy priors. In the validation step, a reliability-based metric is used to evaluate the predictions based on calibrated model parameters and discrepancy in the validation domain. The validation metric serves as a quantitative measure of how well the discrepancy formulation captures the physics missing in the model. In the combination step, the posterior distributions of model parameter and discrepancy corresponding to different priors are combined into a single distribution based on the probabilistic weights derived from the validation step. The combined distribution acknowledges the uncertainty in the prior formulation of model discrepancy function.