This paper reviews the literature on Bayesian experimental design. A unified view of this topic is presented, based on a decision-theoretic approach. This framework casts criteria from the Bayesian literature of design as part of a single coherent approach. The decision-theoretic structure incorporates both linear and nonlinear design problems and it suggests possible new directions to the experimental design problem, motivated by the use of new utility functions. We show that, in some special cases of linear design problems, Bayesian solutions change in a sensible way when the prior distribution and the utility function are modified to allow for the specific structure of the experiment. The decision-theoretic approach also gives a mathematical justification for selecting the appropriate optimality criterion.

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Bayesian Experimental Design: A Review

Subjectivism has become the dominant philosophical foundation for Bayesian inference. Yet in practice, most Bayesian analyses are performed with so-called “noninformative” priors, that is, priors constructed by some formal rule. We review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used. We give special emphasis to Jeffreys's rules and discuss the evolution of his viewpoint about the interpretation of priors, away from unique representation of ignorance toward the notion that they should be chosen by convention. We conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly: When sample sizes are small (relative to the number of parameters being estimated), it is dangerous to put faith in any “default” solution; but when asymptotics take over, Jeffreys's rules and their variants remain reasonable choices. We also provide an annotated b...

The Selection of Prior Distributions by Formal Rules

This chapter provides an overview of recent developments in the design and analysis of cross-over trials. We first consider the analysis of the trial that compares two treatments, A and B, over two periods and where the subjects are randomized to the treatment sequences AB and BA. We make the distinction between fixed and random effects models and show how these models can easily be fitted using modern software. Issues with fitting and testing for a difference in carry-over effects are described and the use of baseline measurements is discussed. Simple methods for testing for a treatment difference when the data are binary are also described. Various designs with two or more treatments but with three or four periods are then described and compared. These include the balanced and partially balanced designs for three or more treatments and designs for factorial treatment combinations. Also described are nearly balanced and nearly strongly balanced designs. Random subject-effects models for the designs with two or more treatments are described and methods for analysing non-normal data are also given. The chapter concludes with a description of the use of cross-over designs in the testing of bioequivalence.

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Design and Analysis of Cross-Over Trials

Maximizing data information requires careful selection, termed design, of the points at which data are observed. Experimental design is reviewed here for broad classes of data collection and analysis problems, including: fractioning techniques based on orthogonal arrays, Latin hypercube designs and their variants for computer experimentation, efficient design for data mining and machine learning applications, and sequential design for active learning. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.

Experimental design

A uniform design (UD) seeks design points that are uniformly scattered on the domain. It has been popular since 1980. A survey of UD is given in the first portion: The fundamental idea and construction method are presented and discussed and examples are given for illustration. It is shown that UD's have many desirable properties for a wide variety of applications. Furthermore, we use the global optimization algorithm, threshold accepting, to generate UD's with low discrepancy. The relationship between uniformity and orthogonality is investigated. It turns out that most UD's obtained here are indeed orthogonal.

Uniform Design: Theory and Application

A general method for obtaining highly efficient factorial designs of relatively small sizes is developed for cDNA microarray experiments. It allows the main effects and interactions to be of possibly unequal importance. First, the approximate theory is employed to get an optimal design measure which is then discretized. It is, however, observed that a naive discretization may fail to yield an exact design of the stipulated size and, even when it yields such an exact design, there is often scope for improvement in efficiency. To address these issues, we propose a step-up/down procedure which is seen to work very well. The resulting designs turn out to be quite robust to possible dye-color effects and heteroscedasticity. We focus on the baseline and all-to-next parametrizations but our method works equally well also for hybrids of the two and other parametrizations.

Highly efficient factorial designs for cDNA microarray experiments: use of approximate theory together with a step-up step-down procedure

On likelihood inference in binary mixed model with an application to COPD data

Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the factor space, is taken as the design criterion. Optimal designs under this criterion are derived, via a combination of algebraic and numerical techniques, for the full second-order regression model over cuboidal regions. Use of a convexity argument and a surrogate objective function significantly reduces the computational burden.

Minimax second-order designs over cuboidal regions for the difference between two estimated responses

We consider a very large class of tests containing in particular the likelihood ratio, the conditional likelihood ratio, Rao's and Wald's tests. In the presence of a nuisance parameter and under contiguous alternatives the third-order power of tests within the class has been studied. A third order optimality property of Rao's test has been proved. The performance of the conditional likelihood ratio test vis-a-vis the usual likelihood ratio test has been investigated. No assumption has been made regarding curved exponentiality.

Rahul Mukerjee

Papers

Highly efficient factorial designs for cDNA microarray experiments: use of approximate theory together with a step-up step-down procedure

On likelihood inference in binary mixed model with an application to COPD data

Minimax second-order designs over cuboidal regions for the difference between two estimated responses

Comparison of tests in the presence of a nuisance parameter

Characterization of normality within the class of elliptical contoured distributions