Bayesian experimental designs for Dubov’s functionals: examples and branching effect
Different approaches used in designs an experiment for nonlinear regression models are locally optimal design, minimax, maximin bayesian and bayesian. All these approaches correspond to different levels of a priori information about the parameters of the model. The bayesian design of construction experimental designs for one-parametric exponential regression model are considered in this article. Five Bayesian Dubov’s D-functionals are used as a optimally criterion. Necessary and sufficient optimal conditions generalizing classic theorem of Kifer-Volfovich to bayesian case are obtained in Grigoriev, Dubov, Fedorov, Atkinson and Donev works for these functionals. In the one-dimensional case, these five functionals reduce to three functionals that generate three classes of bayesian designs. In the general case, the construction of such designs requires the assignment of an a priory distribution satisfying the corresponding regularity conditions. In this paper, a uniform distribution is considered as such a distribution. For the Dubov functionals under consideration and for a given a priori distribution, one-point bayesian designs are constructed and their branch points are found, that is, parameters of the support of the a priori distribution for which the spectrum of the one-point design is replenished by the second point. With a further increase in the diameter of the carrier, the spectrum of the optimal design is expanded due to the branching of the second point of the spectrum, and so on. The problem of finding the second branch point in the paper is not considered. To verify bayesian designs for optimality, the Chaloner-Larnz optimality theorem is used, from which it follows that in the general case the bayesian spectrum may contain a number of points exceeding the number of unknown parameters of the model. In this article for example of one-parameter exponential model shows that three concerned Dubov’s functional (in general case-all five functional) possess different level of self-descriptiveness. For more information criterion being among three mentioned Dubov’s functional, first bifurcation point is taken to infinity. It is mean that bayesian optimal design is one-pointed for any diameter bearer of own a prior distribution.