This paper reviews the literature on Bayesian experimental design, both for linear and nonlinear models. A unified view of the topic is presented by putting experimental design in a decision theoretic framework. This framework justifies many optimality criteria, and opens new possibilities. Various design criteria become part of a single, coherent approach.

This paper explores numerical methods for stochastic optimization, with special attention to Bayesian design problems. A common and challenging situation occurs when the objective function (in Bayesian applications the expected utility) is very expensive to evaluate, perhaps because it requires integration over a space of very large dimensionality. Our goal is to explore a class of optimization algorithms designed to gain efficiency in such situations, by exploiting smoothness of the expected utility surface and borrowing information from neighboring design points. The central idea is that of implementing stochastic optimization by curve fitting of Monte Carlo samples. This is done by simulating 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 optimal design can then be found deterministically. In this paper we introduce a general algorithm for curve-fitting-based optimization, we discuss implementation options, and we present a consistency property for one particular implementation of the algorithm. To illustrate the advantages and limitations of curve-fitting-based optimization, and compare it with some of the alternatives, we consider in detail three important practical applications. The first is an information theoretical stopping rule for a clinical trial. The objective function is based on the expected amount of information acquired about a sub-vector of parameters of interest. The second is concerned with the timing of examination for the early detection of breast cancer in mass screening programs. It involves a two-dimensional optimization and an objective function embodying a cost-benefit analysis. The third applicat...

this paper. We refer to West, Muller and Escobar (1994) and Muller, Erkanli and West (1994). 3 EXAMPLES FOR ONE-STAGE OPTIMAL DESIGNS 3.1 Example 1: An information theoretic stopping rule

Designs for nonlinear regression models depend on some prior information about the unknown parameters. There are three primary methods for accounting for this: The locally optimal designs, globally optimal Bayesian designs, and sequential procedures. If prior knowledge about the parameters is available from former experiments, Bayesian designs integrate this information most efficiently. If the experiments have been performed with replicate measurements, these results can be used to take a heteroscedastic error model into account. We propose a globally optimal, weighted Bayesian design for a heteroscedastic error structure, which can be extended to sequential design procedures. These concepts have been motivated by an application problem in pharmacology: Given a limited number of treatments in bioassay-experiments, an optimal design for dose-response-curves should be determined. 1 INTRODUCTION Bioassays or dose-response-experiments are widely applied in biological and pharmacological...

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We consider maximin and Bayesian D-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior distribution for these parameters is available. It was observed empirically by many authors that an increase of uncertainty in the prior information (i.e. a larger range for the parameter space in the maximin criterion or a larger variance of the prior distribution in the Bayesian criterion) yields a larger number of support points of the corresponding optimal designs. In this paper we present a rigorous proof of this phenomenon and show that in many nonlinear regression models the number of support points of Bayesian- and maximin D-optimal designs can become arbitrarily large if less prior information is available. Our results also explain why maximin D-optimal designs are usually supported at more different points than Bayesian D-optimal designs.