Stirling's approximations for exchangeable Gibbs weights

TL;DR

This paper develops approximation methods for Gibbs partition weights using Stirling numbers, with applications to Bayesian species sampling estimation under inverse Gaussian priors.

Contribution

It introduces new approximation techniques for exchangeable Gibbs weights based on Stirling numbers, enhancing Bayesian nonparametric inference.

Findings

Derived approximation results for Gibbs weights

Applied to Bayesian estimation of discovery probabilities

Improved understanding of partition structures in species sampling

Abstract

We obtain some approximation results for the weights appearing in the exchangeable partition probability function identifying Gibbs partition models of parameter $\alpha \in (0,1)$, as introduced in Gnedin and Pitman (2006). We rely on approximation results for central and non-central generalized Stirling numbers and on known results for conditional and unconditional $\alpha$ diversity. We provide an application to an approximate Bayesian nonparametric estimation of discovery probability in species sampling problems under normalized inverse Gaussian priors.