A Bayesian inference of P(π1 > π2) for two proportions

The statistical inference concerning the difference between two independent binominal proportions is often discussed in medical and statistical literature. However, such discussions are often based on the frequentist viewpoint rather than the Bayesian viewpoint. In this article, we propose an index θ =P(π(1, post ) > π(2, post )), where π(1, post ) and π(2, post ) denote binominal proportions following posterior density. We provide approximate and exact expressions for θ by using the beta prior. We also present the results of actual clinical trials to show the utility of θ. Our findings suggest that θ can potentially provide useful information in a clinical trial.