OPTIMAL BAYESIAN EXPERIMENTAL DESIGN FOR LINEAR MODELS

Optimal experimental designs for linear models have received, and continue to receive, considerable attention in the statistical literature. Whilst this problem has been studied extensively, little is known about optimal designs in a Bayesian framework. In this thesis optimal Bayesian designs for estimation and prediction are derived and discussed for linear models with a prior distribution on the parameters. The designs are optimal for estimating linear combinations c('T)(theta), of the regression coefficients (theta), or for predicting at a point where the expected response is c('T)(theta). The criterion used and developed in this thesis, (psi)-optimality, minimizes the expected squared error loss. We introduce a distribution on c to represent the interest in particular linear combinations of the parameters. This distribution gives rise to the matrix (psi), which is defined to be E(cc('T)). In the usual notation of linear models (psi)-optimality reduces to finding the matrix XX('T) which minimizes tr(psi)(R + XX('T))('-1), where the matrix R is the prior precision matrix of (theta). The designs derived depend not only on the prior distribution and the number of observations to be chosen, but also on what is to be estimated or where prediction is required.;Some general results are given concerning (psi)-optimal designs. In particular, the minimum number of points at which it is necessary to take observations and the conditions under which a one-point design is optimal are given. We also find necessary and sufficient conditions for a design to be (psi)-optimal which lead to some geometric interpretations. Following usual practice the optimal designs derived do not necessarily involve integer numbers of observations. The approximation to integer designs is discussed and it is shown that we expect to lose little by rounding the non-integer designs to integer designs. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of school.) UMI.;In an non-Bayesian context, (psi)-optimal designs are optimal for choosing a design to augment a previously chosen design. The (psi)-optimality criterion is also applicable in certain non-Bayesian problems in experimental design for the estimation of response surfaces.