the influences from model inadequacy and parameter uncertainty, but also the lack of data from both computer simulations and experiments. A common situation is that the input settings are not the same over experiments and simulations. Another common situation is that parameters derived from experiments are input to simulations. For computationally expensive simulations, then often a surrogate model, e.g. a Gaussian process or a Polynomial Chaos Expansion, is necessary, defining an inverse problem for finding the surrogate model that best approximates the simulations. Modular approach An approach to inverse uncertainty quantification is the modular Bayesian approach. The modular Bayesian approach derives its name from its four-module procedure. Apart from the current available data, a prior distribution of unknown parameters should be assigned. Module 1: Gaussian process modeling for the computer model To address the issue from lack of simulation results, the computer model is replaced with a Gaussian process (GP) model where is the dimension of input variables, and is the dimension of unknown parameters. While is pre-defined, , known as hyperparameters of the GP model, need to be estimated via maximum likelihood estimation (MLE). This module can be considered as a generalized kriging method. Module 2: Gaussian process modeling for the discrepancy function Similarly with the first module, the discrepancy function is replaced with a GP model where Together with the prior distribution of unknown parameters, and data from both computer models and experiments, one can derive the maximum likelihood estimates for . At the same time, from Module 1 gets updated as well. Module 3: Posterior distribution of unknown parameters Bayes' theorem is applied to calculate the posterior distribution of the unknown parameters: where includes all the fixed hyperparameters in previous modules. Module 4: Prediction of the experimental response and discrepancy function Full approach Fully Bayesian approach requires that not only the priors for unknown parameters but also the priors for the other hyperparameters should be assigned. It follows the following steps: Derive the posterior distribution ; Integrate out and obtain . This single step accomplishes the calibration; Prediction of the experimental response and discrepancy function. However, the approach has significant drawbacks: For most cases, is a highly intractable function of . Hence the integration becomes very troublesome. Moreover, if priors for the other hyperparameters are not carefully chosen, the complexity in numerical integration increases even more. In the prediction stage, the prediction (which should at least include