Estimating functions evaluated by simulation: A Bayesian/analytic approach

• 成果类型: Article
• 署名作者: Koehler, JR; Puhalskii, AA; Simon, B
• 署名单位: University of Colorado System; University of Colorado Denver
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 1998
• 页码: 1184-1215
• 关键词: computer experiments; prediction

摘要

Consider a function f: B --> R, where B is a compact subset of Rm and consider a simulation used to estimate f(x), x epsilon B with the following properties: the simulation can switch from, one x epsilon B to another in zero time, and a simulation at x lasting t units of time yields a random variable with mean f(x) and variance nu(x)/t. With such a simulation we can divide T units of time into as many separate simulations as we like. Therefore, in principle we can design an experiment that spends tau(A) units Of time simulating points in each A epsilon B, where B is the Borel sigma-field on B and tau is an arbitrary finite measure on (B, B). We call a design specified by a measure tau a generalized design. We propose an approximation for f based on the data from a generalized design. When tau is discrete, the approximation, (f) over cap, reduces to a Kriging-like estimator. We study discrete designs in detail, including asymptotics (as the length of the simulation increases) and a numerical procedure for finding optimal n-point designs based on a Bayesian interpretation of (f) over cap. Our main results, however, concern properties of generalized designs. In particular we give conditions for integrals of (f) over cap to be consistent estimates of the corresponding integrals of f. These conditions are satisfied for a large class of functions, f, even when nu(x) is not known exactly. If f is continuous and tau has a density, then consistent estimation of f(x), x epsilon B is also possible. Finally, we use the Bayesian interpretation of f to derive a variational problem satisfied by globally optimal designs. The variational problem always has a solution and we describe a sequence of n-point designs that approach (with respect to weak convergence) the set of globally optimal designs. Optimal designs are calculated for some generic examples. Our numerical studies strongly suggest that optimal designs have smooth densities.