Several competing objectives may be relevant in the design of an experiment. The competing objectives may not be easy to characterize in a single optimality criterion. One approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria. An alternative approach has been to optimize one criterion subject to constraints on the other criteria. An equivalence theorem is presented for the Bayesian constrained design problem. Equivalence theorems are essential in verifying optimality of proposed designs, especially when, as in most nonlinear design problems, numerical optimization is required. This theorem is used to show that the results of Cook and Wong on the equivalence of the weighted and constrained problems also apply much more generally. The results are applied to Bayesian nonlinear design problems with several objectives. KEY WORDS: Bayesian design, regression, nonlinear design 1. INTRODUCTION An experimen...

Abstract. I describe a framework for adaptive scientific exploration based on iterating an Observation–Inference–Design cycle that allows adjustment of hypotheses and observing protocols in response to the results of observation on-the-fly, as data are gathered. The framework uses a unified Bayesian methodology for the inference and design stages: Bayesian inference to quantify what we have learned from the available data and predict future data, and Bayesian decision theory to identify which new observations would teach us the most. When the goal of the experiment is simply to make inferences, the framework identifies a computationally efficient iterative “maximum entropy sampling ” strategy as the optimal strategy in settings where the noise statistics are independent of signal properties. Results of applying the method to two “toy ” problems with simulated data—measuring the orbit of an extrasolar planet, and locating a hidden one-dimensional object—show the approach can significantly improve observational efficiency in settings that have well-defined nonlinear models. I conclude with a list of open issues that must be addressed to make Bayesian adaptive exploration a practical and reliable tool for optimizing scientific exploration.

... this paper is on illustrating recent computational techniques, and on comparing nonsequential designs with certain sequential alternatives. In particular, the nonsequential design is analyzed in the fixed sample size case, and computed based on an approximation used by Chaloner and Larntz (1989). The sequential design is an heuristic up-down design implemented via a simulation based numerical optimization scheme.

The information theoretic approach to optimal design of experiments yields a simple design criterion: the optimal design minimizes the expected posterior entropy of the parameters. Unfortunately, this strategy is often computational infeasible for non-linear problems and numerical approximations are required. This paper reviews the information theoretic approach to design of experiments, and examines computational issues related to the minimization of the expected posterior entropy and its asymptotic approximations. It is shown that Maximum Entropy Sampling simplifies both the formulation of the design criterion and the optimization problem. Numerical advantages are shown in an example, where the exact solution is compared to asymptotic optimal solution.

this paper we present a general decision theoretic setup for the design problem and develop a solution using Monte Carlo-based methods. Following Kuo (1983), to relax the restrictive assumptions on the potency curve, we adopt a nonparametric Bayesian approach and assume a Dirichlet process prior (Ferguson, 1973) with parameter ffF 0 on the potency curve, where ff represents our strength of prior belief and F 0 is the prior mean of the random tolerance distribution. In particular, we model the potency curve as a random discrete distribution function with random locations and random size of the jumps with the specific distribution given in Sethuraman and Tiwari (1982). The prior mean of this random distribution evaluated at t is assumed to be F 0 (t) and the variance of this random distribution evaluated at t is assumed to be F 0 (t)(1 \Gamma F 0 (t))=(ff + 1). So ff can be interpreted as the strength of prior belief or the degree of concentration of the random distribution around F 0 . The larger the ff is, the more concentrated F is around F 0 . In Section 2, we describe the design problem and introduce its ingredients. We present a decision theoretic setup for the design problem in Section 3 and illustrate the difficulties involved in obtaining an analytical solution to the preposterior analysis. In Section 4 we discuss how MCMC methods can be used for evaluating the posterior expected loss. We present a Monte Carlo integration technique to evaluate the preposterior expected loss and discuss the potential computational burden involved in finding the optimal design. To alleviate such computational burden, we adopt the simulation-based approach of Muller and Parmigiani (1995) where the preposterior analysis is replaced by a curve (surface)-fitting technique. This simulat...

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

... In this paper we discuss implementation strategies for fast numerical computations of Entropies and Kullback-Leibler divergences that are relevant to Bayesian inference and design problems. We illustrate the methods proposed with examples in model diagnostics and information theoretic design.

This paper introduces a numerical method for finding optimal or approximately optimal decision rules and corresponding expected losses in Bayesian sequential decision problems. The method, based on the classical backward induction method, constructs a grid approximation to the expected loss at each decision time, viewed as a function of certain statistics of the posterior distribution of the parameter of interest. In contrast with most existing techniques, this method has a computation time which is linear in the number of stages in the sequential problem. It can also be applied to problems with insufficient statistics for the parameters of interest. Furthermore, it is well-suited to be implemented using parallel processors.

designs for drug development

We consider single and two-period portfolio selection problems for a decision maker with a specified utility function when the variance of security returns is described by a discrete time stochastic model. We present a simulationbased method to solve these problems adopting the approach proposed by Mueller and Parmigiani (1995). This approach replaces the preposterior analysis by a curve fitting based optimization approach. We illustrate the implementation of our approach by an example using an autoregressive conditional heteroskedastic (ARCH) model. Keywords: DYNAMIC PROGRAMMING; PREPOSTERIOR ANALYSIS; DECISION ANALYSIS; STOCHASTIC VOLATILITY. 1.