Estimating functions evaluated by simulation: A Bayesian/analytic approach

Consider a function f: B --> R, where B is a compact subset of Rm and consi der a "simulation" used to estimate f(x), x epsilon B with the following pr operties: the simulation can switch from, one x epsilon B to another in zer o time, and a simulation at x lasting t units of time yields a random varia ble with mean f(x) and variance nu(x)/t. With such a simulation we can divi de 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 tim e simulating points in each A epsilon B, where B is the Borel sigma-field o n B and tau is an arbitrary finite measure on (B, B). We call a design spec ified 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, th e approximation, (f) over cap, reduces to a "Kriging"-like estimator. We st udy discrete designs in detail, including asymptotics (as the length of the simulation increases) and a numerical procedure for finding optimal n-poin t designs based on a Bayesian interpretation of (f) over cap. Our main resu lts, however, concern properties of generalized designs. In particular we g ive conditions for integrals of (f) over cap to be consistent estimates of the corresponding integrals of f. These conditions are satisfied for a larg e class of functions, f, even when nu(x) is not known exactly. If f is cont inuous 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 de rive a variational problem satisfied by globally optimal designs. The varia tional problem always has a solution and we describe a sequence of n-point designs that approach (with respect to weak convergence) the set of globall y optimal designs. Optimal designs are calculated for some generic examples . Our numerical studies strongly suggest that optimal designs have smooth d ensities.