A default Bayesian multiple comparison of two binomial proportions

We consider a default Bayesian approach to multiple test-ing of equality of two binomial proportions. While our ap-proach is motivated by a scenario where one proportion cor-responds to an experimental condition and the other to a control, we find it is also reasonable for comparing two pro-portions in general. We consider a selection of priors un-der the alternative(s) including the intrinsic prior and a newly proposed "mode-based" Beta prior, and investigate their properties in terms of certain desirable characteristics that we specify for default priors. We also develop priors for the hyperparameters based on the conventional hyperprior used for normal means multiple testing. We also consider a computationally more efficient empirical Bayes approach us-ing the intrinsic prior and the proposed Beta prior. We use repeated simulation and real data sets to evaluate and illus-trate the approach, and compare certain frequentist char-acteristics of the results based on intrinsic and mode based Beta prior using full Bayes and empirical Bayes approaches. Additionally, the results from the Bayesian approach are compared with a commonly used frequentist procedure us-ing conventional thresholds in the respective settings. Over-all, we find that the proposed mode-based Beta prior is a suitable default prior for multiple testing of equality of two proportions.