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

Split-plot designs find wide applicability in multifactor experiments with randomization restrictions Practical considerations often warrant the use of unbalanced designs This paper investigates randomization based causal inference in split-plot designs that are possibly unbalanced Extension of ideas from the recently studied balanced case yields an expression for the sampling variance of a treatment contrast estimator as well as a conservative estimator of the sampling variance However, the bias of this variance estimator does not vanish even when the treatment effects are strictly additive A careful and involved matrix analysis is employed to overcome this difficulty, resulting in a new variance estimator, which becomes unbiased under milder conditions A construction procedure that generates such an estimator with minimax bias is proposed

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Causal Inference from Possibly Unbalanced Split-Plot Designs: A Randomization-based Perspective

A one-period inventory situation where the supply is an NBUE random variable with mean proportional to the quantity ordered has been considered. The optimal exponential order quantity, which maximizes the minimum profit obtainable in the NBUE class of supply distributions, is a function of the demand distribution function. Here we show that an estimator of the maximin order quantity, which is already known to converge almost surely to its true value, converges also in distribution to an appropriate normal law with increasing sample size.

Asymptotic normality of the estimated optimal order quantity for one-period inventories with supply uncertainty

Near-factorial experiments in nested row–column designs regulating efficiencies

Second-Order Pitman Admissibility and Pitman Closeness

On the basis of a negative binomial sampling scheme, we consider a uniformly most accurate upper confidence limit for a small but unknown proportion, such as the proportion of defectives in a manufacturing process. The optimal stopping rule, with reference to the twin criteria of the expected length of the confidence interval and the expected sample size, is investigated. The proposed confidence interval has also been compared with several others that have received attention in the recent literature.

Rahul Mukerjee

Papers

Causal Inference from Possibly Unbalanced Split-Plot Designs: A Randomization-based Perspective

Asymptotic normality of the estimated optimal order quantity for one-period inventories with supply uncertainty

Near-factorial experiments in nested row–column designs regulating efficiencies

Second-Order Pitman Admissibility and Pitman Closeness

Interval estimation of a small proportion via inverse sampling