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

Variance and covariance inequalities for truncated joint normal distribution via monotone likelihood ratio and log-concavity

Publisher Summary This chapter focuses on parallel line assays. These are indirect assays with quantitative responses where the experimenter wishes to estimate the relative potency under the assumption that the relationship between the expected response and the logarithm of doses of two preparations is representable by parallel straight lines. In a parallel line assay, the predetermined doses of the standard and test preparations represent the treatments. A parallel line assay is called “symmetric” if the standard and test preparations involve the same number of doses; otherwise, it is called “asymmetric.” The chapter also provides a brief introduction to block designs in the context of parallel line assays along with a discussion of the efficient designing of symmetric parallel line assays. The chapter discusses several open problems associated with parallel line assays.

23 Developments in incomplete block designs for parallel line bioassays

A characterization for orthogonal arrays of strength two via a regression model

It has been shown that under Stein's two-stage sampling procedure, the confidence interval for the population mean, as proposed by him cannot be improved upon. Allied results in the context of point estimation have also been indicated.

Some optimality results on stein's two-stage sampling

With reference to a baseline parametrization, we explore highly efficient, fractional factorial designs for inference on the main effects and, perhaps, some interactions. Our tools include approximate theory together with certain, carefully devised discretization procedures. The robustness of these designs to possible model misspecification is investigated using a minimaxity approach. Examples are given to demonstrate that our technique works well even when the run size is quite small.

Rahul Mukerjee

Papers

Variance and covariance inequalities for truncated joint normal distribution via monotone likelihood ratio and log-concavity

23 Developments in incomplete block designs for parallel line bioassays

A characterization for orthogonal arrays of strength two via a regression model

Some optimality results on stein's two-stage sampling

Approximate theory-aided robust efficient factorial fractions under baseline parametrization