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

Fourth-order rotatable designs: A-optimal measures

We consider two-level fractional factorial designs under a baseline parametrization that arises naturally when each factor has a control or baseline level. While the criterion of minimum aberration can be formulated as usual on the basis of the bias that interactions can cause in the estimation of main effects, its study is hindered by the fact that level permutation of any factor can impact such bias. This poses a serious challenge especially in the practically important highly fractionated situations where the number of factors is large. We address this problem for regular designs via explicit consideration of the principal fraction and its cosets, and obtain certain rank conditions which, in conjunction with the idea of minimum moment aberration, are seen to work well. The role of simple recursive sets is also examined with a view to achieving further simplification. Details on highly fractionated minimum aberration designs having up to 256 runs are provided.

Optimal two-level regular designs under baseline parametrization via Cosets and minimum moment aberration

SUMMARY Optimal designs for diallel crosses are considered when, in addition to the block effects and general combining abilities, the model also includes specific combining abilities. It is seen that partially balanced incomplete block designs with the triangular association scheme play a key role in this context.

Miscellanea. Optimal designs for diallel crosses with specific combining abilities

SUMMARY Conditions are obtained for the approximate frequentist validity of the posterior quantiles of any smooth parametric function. An application to Bayesian tolerance limits is indicated.

Second‐order probability matching priors for a parametric function with application to Bayesian tolerance limits

In a Bayesian setup, we consider the problem of predicting a dependent variable given an independent variable and past observations on the two variables. An asymptotic formula for the relevant posterior predictive density is worked out. Considering posterior quantiles and highest predictive density regions, we then characterize priors that ensure approximate frequentist validity of Bayesian prediction in the above setting. Application to regression models is also discussed.

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Rahul Mukerjee

Papers

Fourth-order rotatable designs: A-optimal measures

Optimal two-level regular designs under baseline parametrization via Cosets and minimum moment aberration

Miscellanea. Optimal designs for diallel crosses with specific combining abilities

Second‐order probability matching priors for a parametric function with application to Bayesian tolerance limits

Probability matching priors for predicting a dependent variable with application to regression models