Bayesian inference for the uncertainty distribution of computer model outputs

We consider a problem of inference for the output of a computationally expensive computer model. We suppose that the model is to be used in a context where the values of one or more inputs are uncertain, so that the input configuration is a random variable. We require to make inference about the induced distribution of the output. This distribution is called the uncertainty distribution, and the general problem is known to users of computer models as uncertainty analysis. To be specific, we develop Bayesian inference for the distribution and density functions of the model output. Modelling the output, as a function of its inputs, as a Gaussian process, we derive expressions for the posterior mean and variance of the distribution and density functions, based on data comprising observed outputs at a sample of input configurations. We show that direct computation of these expressions may encounter numerical difficulties. We-develop an alternative approach based on simulating approximate realisations from the posterior distribution of the output function. Two examples are given to illustrate our methods.