Bayesian Calibration of Expensive Multivariate Computer Experiments

This chapter is concerned with how to calibrate a computer model to observational data when the model produces multivariate output and is expensive to run. The significance of considering models with long run times is that they can be run only 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. In this chapter we use this ensemble to train a meta-model of the computer simulator, which we refer to as an emulator (Sacks et al. 1989). The emulator provides a probabilistic description of our beliefs about the computer model and can be used as a cheap surrogate for the simulator in the calibration process. For any input configuration not in the original ensemble of model runs, the emulator provides a probability distribution describing our uncertainty about the model’s output. The Bayesian approach to calibration of computer experiments using emulators was described by Kennedy and O’Hagan (2001). Their approach was for univariate computer models, and in this chapter we show how those methods can be extended to deal with multivariate models. We use principal component analysis to project the multivariate model output onto a lower dimensional space, and then use Gaussian processes to emulate the map from the input space to the lower dimensional space. We can then reconstruct from the subspace to the original data space. This gives a