29 Review of optimal bayes designs

Abstract

Publisher Summary This chapter focuses on optimal Bayes designs. Proper design of an experiment is evidently a crucial aspect of sound statistical practice; a classic course on design actually helps bring this out much more than a sophisticated course on the mathematics of design. The five most widely accepted criteria for an optimality theory of designs are e, A, D, E, and G optimality. In construction of optimal designs, it is necessary to only consider probability measures resulting in admissible information matrices. In polynomial regression problems, because of the moment interpretation of the information matrix, this helps in bounding the number of support points in an optimal design according to any criterion that is monotone increasing in the moment matrix in the Loewner ordering. The mathematics of Bayes optimal designs is generally the same as that in classical optimal design. There are three main routes to obtaining an optimal design: (1) by using an equivalence theorem, (2) in polynomial models, by using inherent symmetry in the problem (if there is such symmetry) and convexity of the criterion functional in conjunction with Caratheodory type bounds on the cardinality of the support, and (3) by using geometric arguments, which usually go by the name of “Elfving geometry.”