Estimating functions evaluated by simulation: a Bayesian/analytic approach

Abstract

Consider a function $f: B \to R_'$, where B is a compact subset of $R^m$ 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 $v(x)/t$ . 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 $A \epsilon \mathscr{B}$, where $\mathscr{B}$ is the Borel $\sigma$-field on B and $\tau$ is an arbitrary finite measure on $(B, \mathscr{B})$. 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, $\hat{f}$, 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 $\hat{f}$ . Our main results, however, concern properties of generalized designs. In particular, we give conditions for integrals of $\hat{f}$ to be consistent estimates of the corresponding integrals of f. These conditions are satisfied for a large class of functions, f , even when $v(x)$ 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 $\hat{f}$ 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.