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...roximation is: `jy; jsN( `; [R + nI( `; j)] \Gamma1 ) (13) where ` now denotes the mode of the joint posterior distribution of ` (also called the gener27 alized maximum likelihood estimate of ` as in =-=Berger, 1985-=-, p.133), and R is the matrix of second derivatives of the logarithm of the prior density function, or the precision matrix of the prior distribution. Several other approximations are possible, for ex...

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...variate t distributions. Denote by t ffi [m; ; \Sigma] the probability distribution of an m-variate t random variable with ffi degrees of freedom, mean vectorsand scale matrix \Sigma (see for example =-=DeGroot 1970-=-, sec 5.6 or Box and Tiao 1973 page 117). Recall that ` = (nM(j) +R) \Gamma1 (X T y + R` 0 ). Let h(j; y) denote the quantity (2ff+n) \Gamma1 n (y \Gamma X` 0 ) T h I \Gamma X(nM(j) +R) \Gamma1 X T i ...

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...Denote by t ffi [m; ; \Sigma] the probability distribution of an m-variate t random variable with ffi degrees of freedom, mean vectorsand scale matrix \Sigma (see for example DeGroot 1970, sec 5.6 or =-=Box and Tiao 1973-=- page 117). Recall that ` = (nM(j) +R) \Gamma1 (X T y + R` 0 ). Let h(j; y) denote the quantity (2ff+n) \Gamma1 n (y \Gamma X` 0 ) T h I \Gamma X(nM(j) +R) \Gamma1 X T i (y \Gamma X` 0 ) + 2fi o and l...

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...usual, a linear approximation to some smooth but unknown surface, then it is at the boundary of this region that the approximation is most inaccurate. 53 These criticisms are not new (see for example =-=Box and Draper, 1959-=-, Sacks and Ylvisaker, 1984, 1985) and apply to both Bayesian and non-Bayesian optimal design for linear models. As discussed in Section 5.2 these criticisms sometimes do not apply to Bayesian optimal...

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...ion. When inference about the parameters is the main goal of the analysis, for example, a utility function based on Shannon information leads to Bayesian D-optimality in the normal linear model (see, =-=Bernardo, 1979-=-). In addition, Shannon information can be used for prediction and in mixed utility functions that describe several simultaneous goals for an experiment. Bayesian equivalents of some other popular opt...

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...r, as they do not allow for checking of the model after the experiment is performed. The bound also applies to most local optimality criteria and Bayesian criteria for linear models (see, for example =-=Chernoff, 1972-=-, p. 27 and Chaloner, 1984). In contrast for nonlinear models there is no such bound available on the number of support points. Although the criteria are concave on H, the space of probability measure...

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...cess and an informative p(`) will be used to average over in the integral. This echoes the idea given in Tsutakawa (1972) of using different prior distributions for design and for analysis. (See also =-=Etzione and Kadane, 1993-=-). ffl for similar reasons these criteria are appealing in a non-Bayesian framework where it is accepted that prior information must be used in design but should not be used in the analysis. Indeed th...

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...se prior information, this criterion is the same as non-Bayesian D-optimality for testing the hypothesis ` 0 = 0, where ` 0 is the subvector of extra parameters in the larger model (see, for example, =-=Atkinson 1972-=-). For the dual purpose of model discrimination and parameter estimation for two nested normal linear models, Spezzaferri showed that the optimality criterion using utility (5) is given by the product...

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...ian Alphabetical Optimality: Related Work. In the 1970's Lindley's work had a profound influence on many aspects of Bayesian statistics. In the area of experimental design, a set of papers by Brooks (=-=Brooks 1972-=-, 1974, 1976, 1977) were inspired by work of Lindley's on the choice of variables in multiple regression (Lindley 1968). Brooks followed Lindley's approach to motivate the problem of choosing the best...

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...y easy but, as is shown in Section 2.5, algebraically hard. Pilz dealt with Bayes experimental designs for a linear model in a series of papers (Pilz 1979a, b, c, d, 1981a, b, c Nather and Pilz 1980, =-=Gladitz and Pilz 1982-=-a, b, Bandemer, Nather and Pilz 1987). See also the monograph, Pilz (1983) and the revised reprint of the monograph, Pilz (1991). His approach is very general, with no distributional assumptions for t...

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... used. The choice of a utility (or loss) function expresses various reasons for carrying out an experiment. In the linear model, the analogs of widely known non-Bayesian alphabetical design criteria (=-=Box, 1982-=-) such as A-optimality, D-optimality and others can be given decision theoretic justification. When inference about the parameters is the main goal of the analysis, for example, a utility function bas...

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...y easy but, as is shown in Section 2.5, algebraically hard. Pilz dealt with Bayes experimental designs for a linear model in a series of papers (Pilz 1979a, b, c, d, 1981a, b, c Nather and Pilz 1980, =-=Gladitz and Pilz 1982-=-a, b, Bandemer, Nather and Pilz 1987). See also the monograph, Pilz (1983) and the revised reprint of the monograph, Pilz (1991). His approach is very general, with no distributional assumptions for t...