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A0524
Title: Discrete nonparametric priors with fixed mean distributions Authors:  Francesco Gaffi - University of Notre Dame (United States) [presenting]
Antonio Lijoi - Bocconi University (Italy)
Igor Pruenster - Bocconi University (Italy)
Abstract: Functionals of random probability measures are objects of great interest from a probabilistic perspective. They also play an important role in Bayesian Nonparametrics. In the latter context, understanding the behavior of a finite-dimensional feature of a flexible and infinite-dimensional prior is crucial for prior elicitation. The classical line of research resorts to the Cifarelli-Regazzini identity and its extensions to determine the distribution of the mean of random probability measures, firstly in the Dirichlet case and then, in greater generality, for the Pitman-Yor process and normalized random measures. This presentation targets the inverse path: determining the (unique) parameter measure of a discrete random probability measure giving rise to a desired mean distribution. Taking this direction yields a better understanding of the set of mean distributions of notable nonparametric priors, giving moreover a way to enforce prior information directly. Such a task has been completed just in the Dirichlet case with a unit concentration parameter for solving a mostly unrelated problem in combinatorics. We provide results for the general Dirichlet case, the normalized stable and the Pitman-Yor processes, with an application to mixture models.