Random probability measures with fixed mean distributions

Linear functionals, or means, of discrete random probability measures are a natural probabilistic object and the investigation of their properties have a long and rich history. They appear in several areas of mathematics, including statistics, combinatorics, special functions, excursions of stochastic processes and financial mathematics, among others. Most contributions have aimed at determining their distribution starting from a fully specified random probability. This work addresses the inverse problem: the identification of the base measure of a discrete random probability measure yielding a specific mean distribution. Available results concern only the Dirichlet case for specific choices of the concentration parameter. Here we address the problem in much greater generality and, besides considering new instances of Dirichlet processes, we cover the normalized stable process and the Pitman–Yor process. In addition to their theoretical interest, the results are of practical relevance to Bayesian nonparametric inference, where the law of a random probability measure acts as a prior distribution: often pre-experimental information is available about a finite-dimensional projection of the data generating distribution, such as the mean, rather than about an infinite-dimensional parameter. We further extend our findings to mixture models, ubiquitous in statistics and machine learning.