Enhanced Objective Bayesian Testing for the Equality of two Proportions

We develop a new class of prior distributions for Bayesian comparison of nested models, which we call intrinsic moment priors, by combining the well-established notion of intrinsic prior with the recently introduced idea of non-local priors, and in particular of moment priors. Specifically, we aim at testing the equality of two proportions, based on independent samples, and thus focus on discrete data models. Given two nested models, each equipped with a default prior, we first construct a moment prior under the larger model. In this way, the asymptotic learning behavior of the Bayes factor is strengthened, relative to currently used local priors, when the smaller model holds; remarkably, this effect is already apparent for moderate sample sizes. On the other hand, the asymptotic learning behavior of the Bayes factor when the larger model holds is unchanged. However, without appropriate tuning, a moment prior does not provide enough evidence for the larger model when the sample size is small and the data only moderately support the smaller one. For this reason, we apply to the moment prior an intrinsic prior procedure, which amounts to pulling the moment prior towards the subspace specified by the smaller model; we provide general guidelines for determining the training sample size necessary to implement this step. Thus, by joining the virtues of moment and intrinsic priors, we obtain an enhanced objective Bayesian testing procedure: i) our evidence for small samples is broadly comparable to that given by current objective methods; ii) we achieve a superior learning performance as the sample size increases (when the smaller model holds). We first illustrate our methodology in a running Bernoulli example, where we test a sharp null hypothesis, then we implement our procedure to test the equality of two proportions. A detailed analysis of the properties of our method, including a comparison with standard intrinsic priors, is presented together with an application to a collection of real-world 2 × 2 tables involving a sensitivity analysis and a crossvalidation study.