Bayesian model comparison based on Wasserstein distances

Exchangeable processes are extensively used in Bayesian nonparametrics to model exchangeable data. Most common approaches assign a law to the process through the specification of a random measure. When two processes only differ in the law of the random measure, a distance between random measures provides a natural way to compare them. In this work we propose one by relying on the Wasserstein distance. Moreover, we overcome the analytical difficulties of evaluating the distance by developing sharp upper and lower bounds. The specialization of these bounds to Gamma random measures provides the exact value of the Wasserstein distance in terms of the Kolmogorov distance between the base measures. The results are based on a forthcoming work in collaboration with A. Lijoi and I. Prunster.