Merging Rate of Opinions via Optimal Transport on Random Measures

The Dirichlet process has been pivotal to the development of Bayesian nonparametrics, allowing one to learn the law of the observations through closed-form expressions. Still, its learning mechanism is often too simplistic and many generalizations have been proposed to increase its flexibility, a popular one being the class of normalized completely random measures. Here we investigate a simple yet fundamental matter: will a different prior actually guarantee a different learning outcome? To this end, we develop a new framework for assessing the merging rate of opinions based on three leading pillars: i) the investigation of identifiability of completely random measures; ii) the measurement of their discrepancy through a novel optimal transport distance; iii) the establishment of general techniques to conduct posterior analyses, unravelling both finite-sample and asymptotic behaviour of the distance as the number of observations grows. Our findings provide neat and interpretable insights on the impact of popular Bayesian nonparametric priors, avoiding the usual restrictive assumptions on the data-generating process.