Specification of the base measure of nonparametric priors via random means

Functionals of random probability measures are probabilistic objects whose properties are studied in different fields. They also play an important role in Bayesian Nonparametrics: understanding the behavior of a finite dimensional feature of a flexible and infinite-dimensional prior is crucial for prior elicitation. In particular distributions of means of nonparametric priors have been the object of thorough investigation in the literature. We target the inverse path: the determination of the parameter measure of a random probability measure giving rise to a fixed mean distribution. This direction yields a better understanding of the sets of mean distributions of notable nonparametric priors, giving moreover a way to directly enforce prior information, without losing inferential power. Here we summarize and report results obtained in [6] for the Dirichlet process, the normalized stable random measure and the Pitman–Yor process, with an application to mixture models.