  Motivated by the fundamental problem of measuring species diversity, this
paper introduces the concept of a cluster structure to define an exchangeable
cluster probability function that governs the joint distribution of a random
count and its exchangeable random partitions. A cluster structure, naturally
arising from a completely random measure mixed Poisson process, allows the
probability distribution of the random partitions of a subset of a sample to be
dependent on the sample size, a distinct and motivated feature that differs it
from a partition structure. A generalized negative binomial process model is
proposed to generate a cluster structure, where in the prior the number of
clusters is finite and Poisson distributed, and the cluster sizes follow a
truncated negative binomial distribution. We construct a nonparametric Bayesian
estimator of Simpson's index of diversity under the generalized negative
binomial process. We illustrate our results through the analysis of two real
sequencing count datasets.
