In this project, we propose to consider problems in Bayesian asymptotics, design, inference, and the question of synthesis with classical procedures. In asymptotics, we propose to consider the limiting behavior of the posterior in nonregular problems. Design of experiments will be considered in nonlinear models, when prior information is available about parameters of the model. Such nonlinear models typically are hard to deal with because of dependence of the design on the parameters themselves. In particular, we will try to use the rich linear theory by approximating nonlinear response functions by linear functions, such as polynomials. We also will derive designs that give a preexperimental guarantee of pos data accuracy, uniformly in the data whenever possible. Finally, in canonical Exponential family problems, we will derive priors for which the implied Bayes procedure has classically satisfactory property. Much of existing statistical methodology has evolved from methods of so called classical statistics. In comparison, the alternative methods known as Bayesian methods are still growing. Nevertheless, it is now recognized by the community as a whole that Bayesian methods can be very useful as tools for statistical data analysis. In this project, our goal is two fold: first, to study alternative Bayesian methods in important classes of statistical problems, such as designing statistical surveys, and second, to bridge the two methods of statistical analysis together by studying Bayesian methods which are also acceptable according to classical criteria.
