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Gibbs partition models are the largest class of infinite exchangeablepartitions of the positive integers generalizing the product form of theprobability function of the two-parameter Poisson-Dirichlet family. Recentlythose models have been investigated in a Bayesian nonparametric approach tospecies sampling problems as alternatives to the Dirichlet and the Pitman-Yorprocess priors. Here we derive marginals of conditional and unconditionalmultivariate distributions arising from exchangeable Gibbs partitions to obtainexplicit formulas for joint falling factorial moments of correspondingconditional and unconditional Gibbs sampling formulas. Our proofs rely on aknown result on factorial moments of sum of non independent indicators. Weprovide an application to a Bayesian nonparametric estimation of the predictiveprobability to observe a species already observed a certain number of times.

Marginals of multivariate Gibbs distributions with applications in Bayesian species sampling