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

In an order-of-addition experiment, each treatment is a permutation of m components. It is often unaffordable to test all the m! treatments, and the design problem arises. We consider a model that incorporates the order of each pair of components and can also account for the distance between the two components in every such pair. Under this model, the optimality of the uniform design measure is established, via the approximate theory, for a broad range of criteria. Coupled with an eigen-analysis, this result serves as a benchmark that paves the way for assessing the efficiency and robustness of any exact design. The closed-form construction of a class of robust optimal fractional designs is then explored and illustrated.

Design of order-of-addition experiments

SUMMARY This paper characterizes priors under which the Bayesian and frequentist Bartlett corrections for the likelihood ratio and the conditional likelihood ratio (CLR) statistics differ by o(1). It is seen that, except for sample points with negligible probability, the CLR statistic has a posterior distribution for which a posterior Bartlett correction exists. This observation leads to an alternative proof of the existence of a frequentist Bartlett correction for the CLR statistic.

Bayesian and frequentist Bartlett corrections for likelihood ratio and conditional likelihood ratio tests

SUMMARY With reference to a general class of empirical-type likelihoods, we develop higher-order asymptotics for the frequentist coverage of Bayesian credible sets based on posterior quantiles and highest posterior density. These asymptotics, in turn, characterise members of the class that allow approximate frequentist validity of such sets. It is seen that the usual empirical likelihood does not enjoy this property up to the order of approximation considered here.

Empirical-type likelihoods allowing posterior credible sets with frequentist validity: Higher-order asymptotics

Even in the absence of blocks, each cross in a partial diallel cross design can be formally identified with a 'block' of size two and thus the design itself can be identified with a binary block design. We show that unblocked partial diallel crosses, so linked with a certain class of group divisible designs, are uniquely E-optimal in general and also uniquely D- and A-optimal in the saturated case. Further results on E-optimality, applicable even when the number of parental lines does not admit a factorisation, are also derived. The issue of optimal or efficient blocking of partial diallel crosses is discussed.

Optimal partial diallel crosses

Priors ensuring frequentist validity, up to o(n ), of credible regions based on the highest posterior density have been characterized in the presence of nuisance parameters. In this connection, the consequences of an orthogonal parametrization have also been discussed. 1: Introduction In recent years, there has been a revival of interest in problems relating to approximate frequentist validity of posterior credible regions. As noted in Tibshirani (1989), apart from providing a method for constructing accurate frequentist confidence regions, such studies are also helpful in defining noninformative priors which could be potentially useful for comparative purposes in a Bayesian analysis. The results available in the literature in this general area include those related to the approximate frequentist validity of one-sided posterior regions based on posterior quantiles ([18], [13], [16], [17], [12]), posterior regions based on the inversion of likelihood ratio and related statistics ([7], [8]) and highest posterior density (HPD) regions ([14], [9]). We refer to [11] and [15] for further interesting results and references. As for the problem of characterizing priors ensuring approximate frequentist validity of HPD regions, it appears that not much work has as yet been done on models involving nuisance parameters. In consideration of the current interest on such models, in this work we propose to fill up this gap to some extent. As a special case, models with an orthogonal parametrization ([5]) have been considered. An advantage of our approach is that it does not require an explicit evaluation of all the coefficients involved in the (approximate) posterior density of the interest parameter and this keeps the algebra relatively simple (see Section 3). AMS subject classification: 62F15, 62F25, 62E20 Key word· and phrases: highest posterior density, non-informative prior, parametric orthogonality.

Rahul Mukerjee

Papers

Design of order-of-addition experiments

Bayesian and frequentist Bartlett corrections for likelihood ratio and conditional likelihood ratio tests

Empirical-type likelihoods allowing posterior credible sets with frequentist validity: Higher-order asymptotics

Optimal partial diallel crosses

Frequentist validity of highest posterior density regions in the presence of nuisance parameters