Approximating predictive probabilities of Gibbs-type priors

TL;DR

This paper demonstrates that predictive probabilities of Gibbs-type priors can be approximated for large sample sizes with negligible error, preserving key features of the well-understood Poisson-Dirichlet prior.

Contribution

It provides a large-sample approximation for the predictive probabilities of any Gibbs-type prior, extending the tractability of the Poisson-Dirichlet case.

Findings

Predictive probabilities admit a large n approximation with o(1/n) error

Approximation maintains desirable features of the Poisson-Dirichlet prior

Enhances understanding of Gibbs-type priors in large-sample settings

Abstract

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most "natural" generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson-Dirichlet prior certainly stands out for the mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in several applications. Given a sample of size $n$, in this paper we show that the predictive probabilities of any Gibbs-type prior admit a large $n$ approximation, with an error term vanishing as $o(1/n)$, which maintains the same desirable features as the predictive probabilities of the two parameter Poisson-Dirichlet prior.