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Keywords:

  • Bayesian non-parametrics;
  • completely random measures;
  • coupling from the past method;
  • Dirichlet process;
  • Gibbs-type random probability measures;
  • normalized random measures with independent increments;
  • predictive distributions;
  • species sampling problems

Abstract.  In this article, we define and investigate a novel class of non-parametric prior distributions, termed the class inline image. Such class of priors is dense with respect to the homogeneous normalized random measures with independent increments and it is characterized by a richer predictive structure than those arising from other widely used priors. Our interest in the class inline image is mainly motivated by Bayesian non-parametric analysis of some species sampling problems concerning the evaluation of the species relative abundances in a population. We study both the probability distribution of the number of species present in a sample and the probability of discovering a new species conditionally on an observed sample. Finally, by using the coupling from the past method, we provide an exact sampling scheme for the system of predictive distributions characterizing the class inline image.