Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments

Abstract

This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” {t (1), t (2), …, t (n)} in T, and with the design, that is, with the selection of those sites. The motivating application is the design and analysis of computer experiments, where t determines the input to a computer model of a physical or behavioral system, and y(t) is a response that is part of the output or is calculated from it. Following a Bayesian formulation, prior uncertainty about the function y is expressed by means of a random function Y, which is taken here to be a Gaussian stochastic process. The mean of the posterior process can be used as the prediction function ŷ(t), and the variance can be used as a measure of uncertainty. This kind of approach has been used previously in Bayesian interpolation and is strongly related to the kriging methods used in geostatistics. Here emphasis is placed on product linear and product cubic correlation functions, which yield prediction functions that are, respectively, linear or cubic splines in every dimension. A posterior entropy criterion is adopted for design; this minimizes the expected uncertainty about the posterior process, as measured by the entropy. A computational algorithm for finding entropy-optimal designs on multidimensional grids is described. Several examples are presented, including a two-dimensional experiment on a computer model of a thermal energy storage device and a six-dimensional experiment on an integrated circuit simulator. Predictions are made using several different families of correlation functions, with parameters chosen to maximize the likelihood. For comparison, predictions are also made via least squares fitting of various polynomial and spline models. The Bayesian design/prediction methods, which do not require any modeling of y, produce comparatively good predictions. For some correlation functions, however, the 95% posterior probability intervals do not give adequate coverage of the true values of y at selected test sites. These methods are fairly simple and offer considerable potential for virtually automatic implementation, although further development is needed before they can be applied routinely in practice.