Bayesian Calibration of Expensive Multivariate Computer Experiments

This chapter is concered with how to calibrate a computer model to observational data when the model produces multivariate output and is temporally expensive to run. The significance of considering models with long run times is that they can only be run at a limited number of different inputs, ruling out a brute-force Monte Carlo approach. Consequently, all inference must be done with a limited ensemble of model runs. A probabilistic approach is taken here, with the aim being to find a probability distribution which represents our uncertainty about the true model inputs given the observational data and the computer model. We assume statistical models for the measurement errors on the observed data and for the discrepancy between the model and reality. We also describe a statistical model for our uncertainty about the computer model’s value at untried input values. We take a Bayesian approach and describe our prior beliefs about the model and update these beliefs after observing an ensemble of model runs. Gaussian process priors are used as a flexible semi-parametric family to describe our beliefs, and the posterior distribution of the process can be considered as a meta-model of the computer model. We refer to the meta-model as an emulator of the computer simulator (Sacks et al. 1989). This approach allows beliefs to be described about the model output at input configurations not in the original design.