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

The earliest important papers on planned experiments in Biometrika appeared within a year of each other in 1917 and 1918. Roughly speaking, one was concerned with design optimality and industrial experiments, the other with agricultural trials and blocking. We find this approximate division of the subject into two parts helpful in describing its development and growth. As a result of the hostility between Fisher and Karl Pearson, much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of Biometrika. Despite this, we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and Bayesian versions of design optimality.

One hundred years of the design of experiments on and off the pages of Biometrika

(k, l) [1885]. (n− k + 1) [1753, 1623]. 0 [772]. 0 < ρ < ρ0 [2026]. 1 [772, 548]. $109.95 [2218]. $125.00 [2161]. $129.95 [2259]. 1 ≤ p <∞ [562]. 2 [616, 407]. $229.00 [2019]. 24 [1398]. 24 [1476]. 2s−k [2039]. 3 [852, 2125]. $49.95 [2161]. $59.95 [2219]. $89.95 [2217]. (R) [2002, 1962]. 1 [1276]. 2 [1417]. k [798]. A [1577, 2312, 2346]. α [858, 1840, 2242, 1465, 1271, 1945, 2327, 1926]. AR(1) [1650, 790]. AR(p) [1346]. R̄ [652]. β [668, 856, 497]. c [2332]. χ [919]. D [2135, 2268, 836, 948, 1324, 2309, 2346, 2098, 2176]. δ [1959]. e [596]. E(x⊗ xx′) [1805]. E(xx′ ⊗ xx′) [1805]. F [1901, 915, 1361, 407, 681]. G [548]. Γ [722]. H [1664]. I(1) [845]. K [2308, 1551, 1223, 1837, 368, 479, 2177, 1322, 1198, 2224, 2309, 407, 1932]. k − 2 [1781]. L [439, 1974, 1584, 1399, 1322]. L1 [895]. L2 [2195, 2363, 2039]. LP [562]. LM [1343]. LR [1343]. M [1054, 1107, 2094, 2180]. Lp [1552]. N [816, 752, 48, 1753, 1623, 548]. p [1245, 1461, 2378, 1794, 1868, 1604, 1791, 1589, 2074, 2227]. P (X < Y ) [1867, 1492, 1955]. P (Y < X) [2222, 1380]. Φ [1345]. Pr(a < x < b) [263]. q [1651]. R [1238, 1848, 616, 1932]. r − k [2082, 1480]. R/S [1996]. R = Pr(X > Y ) [2007]. r [1446]. ρ∗ [1884]. S [2151, 1105]. S [848, 790, 909]. σ [1141, 684]. SUNn,2 [2046]. t [1054, 1777, 327, 1533, 1722, 982, 856, 1031, 1780, 1559, 1094, 1391, 1342, 1680]. T 2

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A Complete Bibliography of Publications in Statistical Papers

The variability among data may have quite complex structure and the splitting of variation into its individual components is important in many areas of applications. In particular, the study of variation through a class of linear models known as random and mixed models is a central topic in statistics with wide ramifications in both theory and applications. Sometimes these models are called components of variance models since the interest in the model lies in investigating the variances of the effects of factor levels which are assumed to be randomly selected from a large population of levels. These variances are also known as variance components since the total variance can be expressed as the sum of these variances and the error variance. Knowledge of the variance components, i.e., the information on the relative importance and magnitude of the sources of variability in the data is of critical importance in many problems since the isolation of the total variance into the individual components of variance would result in the physical elimination of the sources of variability in question.

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A bibliography on variance components an introduction and an update: 1984-2002

Summary This article provides (1) a comprehensive coverage of the literature on designs for estimating variance components, and (2) a review of recent applications of such designs in genetics, statistical process control, and quality improvement. In addition, recent methods of estimation of variance components and model forms, other than the linear, are discussed. The latter developments have a direct effect on the choice of design. Some suggested ideas for future research directions are also given.

Designs for Variance Components Estimation: Past and Present

The design problem for the estimation of variance components by the method of unweighted squares of means, under a multifactor random effects model, is considered. First it is shown that with the numbers of levels of the factors fixed, a balanced design, if it exists, is optimal. Next the numbers of levels are also treated as decision variables and the derivation of minimax designs is indicated.

Optimal design for the estimation of variance components

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

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.

Approximate theory-aided robust efficient factorial fractions under baseline parametrization

A second-order model involving the intercept and only the pure quadratic terms is considered for regression over hypercubes. Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the region of interest in factor space, is taken as the design criterion. Optimal design under the minimax criterion is derived and found to be the one which is also simultaneously A-, D-, and E-optimal for the parameters excluding the intercept. The minimax design is compared with other standard designs and is found to perform extremely well.

Minimax Design for the Difference Between Estimated Responses for the Quadratic Model Over Hypercubic Regions

Approximate theory employing FBECHET derivative is utilized to derive optimal weighing designs under D- and A-optimality criteria. Both spring and chemical balance designs, without and with restriction on the number of objects that may be includ¬ed in a weighing, are considered. The optimality of some exact (discrete) designs is demonstrated

S. Huda

Papers

Optimal design for the estimation of variance components

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

Approximate theory-aided robust efficient factorial fractions under baseline parametrization

Minimax Design for the Difference Between Estimated Responses for the Quadratic Model Over Hypercubic Regions

Optimal weighing designs: approximate theory