A Bayesian Inference and Stochastic Dynamic Programming Approach to Determine the Best Binomial Distribution

In this research, we employ Bayesian inference and stochastic dynamic programming approaches to select the binomial population with the largest probability of success from n independent Bernoulli populations based upon the sample information. To do this, we first define a probability measure called belief for the event of selecting the best population. Second, we explain the way to model the selection problem using Bayesian inference. Third, we clarify the model by which we improve the beliefs and prove that it converges to select the best population. In this iterative approach, we update the beliefs by taking new observations on the populations under study. This is performed using Bayesian rule and prior beliefs. Fourth, we model the problem of making the decision in a predetermined number of decision stages using the stochastic dynamic programming approach. Finally, in order to understand and to evaluate the proposed methodology, we provide two numerical examples and a comparison study by simulation. The results of the comparison study show that the proposed method performs better than that of Levin and Robbins (1981 Levin , B. , Robbins , H. ( 1981 ). Selecting the highest probability in Binomial or multinomial trials . Proc. Nat. Acad. Sci. USA 78 : 4663 – 4666 .Crossref], PubMed], Web of Science ®] , Google Scholar]) for some values of estimated probability of making a correct selection.