Stick-breaking Pitman-Yor processes given the species sampling size

Random discrete distributions, say $F,$ known as species sampling models, represent a rich class of models for classification and clustering, in Bayesian statistics and machine learning. They also arise in various areas of probability and its applications. Jim Pitman, within the species sampling context, shows that mixed Poisson processes may be interpreted as the sample size up till a given time or in terms of waiting times of appearance of individuals to be classified. He notes connections to some recent work in the Bayesian statistic/machine learning literature, with some more classical results. We let $F:=F_{\alpha,\theta},$ be a Pitman-Yor process for $\alpha\in (0,1),$ and $\theta>-\alpha,$ with $\alpha$-diversity equivalent in distribution to $S^{-\alpha}_{\alpha,\theta},$ and let $(N_{S_{\alpha,\theta}}(\lambda),\lambda\ge 0)$ denote a mixed Poisson process with rate $S_{\alpha,\theta}.$ In this paper we derive explicit stick-breaking representations of $F_{\alpha,\theta}$ given $N_{S_{\alpha,\theta}}(\lambda)=m.$ More precisely, if $(P_{\ell})\sim \mathrm{PD}(\alpha,\theta)$, denotes a ranked sequence following the two parameter Poisson-Dirichlet distribution, we obtain explicit representations of the sized biased permutation of $(P_{\ell})|N_{S_{\alpha,\theta}}(\lambda)=m.$ Due to distributional results we shall develop in a more general context, it suffices to consider the stable case $F_{\alpha,0}|N_{S_{\alpha}}(\lambda)=m.$ Notably, it follows that $F_{\alpha,0}|N_{S_{\alpha}}(\lambda)=0,$ is equivalent in distribution to the popular normalized generalized gamma process. Hence, we obtain explicit stick-breaking representations for the generalized gamma class as a special case.