OPTIMAL-DESIGN VIA CURVE-FITTING OF MONTE-CARLO EXPERIMENTS

This article explores numerical methods for stochastic optimization, w ith special attention to Bayesian design problems. A common and challe nging situation occurs when the objective function (in Bayesian applic ations, the expected utility) is very expensive to evaluate, perhaps b ecause it requires integration over a space of very large dimensionali ty. Our goal is to explore a class of optimization algorithms designed to gain efficiency in such situations, by exploiting smoothness of th e expected utility surface and borrowing information from neighboring design points. The central idea is that of implementing stochastic opt imization by curve fitting of Monte Carlo samples. This is done by sim ulating draws from the joint parameter/sample space and evaluating the observed utilities. Fitting a smooth surface through these simulated points serves as estimate for the expected utility surface. The optima l design can then be found deterministically. In this article we intro duce a general algorithm for curve-fitting-based optimization, discuss implementation options, and present a consistency property for one pa rticular implementation of the algorithm. To illustrate the advantages and limitations of curve-fitting-based optimization, and to compare i t with some of the alternatives, we consider in detail two important p ractical applications: an information theoretical stopping rule for a clinical trial, with an objective function based on the expected amoun t of information acquired about a subvector of parameters of interest, and the design of exploratory shock levels in the implantation of hea rt defibrillators. This latter example is also used for comparison wit h some of the alternative schemes. One of the main attractions of effi cient optimization algorithms in design is the application to sequenti al problems. We conclude with-an outlook on how the ideas presented he re can be extended to solve stochastic dynamic programming problems su ch as those occurring in Bayesian sequential design.