Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling

Abstract

Gibbs partition models are the largest class of infinite exchangeable partitions of the positive integers generalizing the product form of the probability function of the two-parameter Poisson-Dirichlet family. Here we call into question the current approach to Bayesian nonparametric estimation in species sampling problems under Gibbs priors, which incorrectly relies on treating exchangeable partition probability functions (EPPFs) as multivariate distributions on compositions of the positive integers. We show that once those multivariate distributions are correctly derived, results for corresponding sampling formulas can be obtained, generalized and sometimes fixed, working with marginals and a known result on falling factorial moments of a sum of non independent indicators. We provide an application of our findings to a recently proposed Bayesian nonparametric estimation under Gibbs priors of the predictive probability to observe a species already observed a certain number of times.