Posterior Predictive Comparisons for the Two-sample Problem

The two-sample problem of inferring whether two random samples have equal underlying distributions is formulated within the Bayesian framework as a comparison of two posterior predictive inferences rather than as a problem of model selection. The suggested approach is argued to be particularly advantageous in problems where the objective is to evaluate evidence in support of equality, along with being robust to the priors used and being capable of handling improper priors. Our approach is contrasted with the Bayes factor in a normal setting and finally, an additional example is considered where the observed samples are realizations of Markov chains.