Renewal theory and the sequential design of experiments with two states of nature

In the sequential design of experiments in experimentel is sequentially performing experiments to help him make an inference about the true state of nature. Using results from renewal twenty at derive approximations for the operations characteristics and average sample numbers for this problem when there are two states of nature. A critical problem in the sequential design of experiments is finding a good procedure. We investigate a Bayesian formulation of this problem and use our approximations to approximate the Bayes risk. Minimization of this approximate Bayes risk over procedures is discussed as a method of finding a good procedure, but difficulties are encountered due to the discrete time character of the sequential process. To avoid these difficulties, we consider minimization of an approximation related to a diffusion process. This leads to a simple rule for the sequential selection of experiments.