Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact conditional finite-mixture representation by augmenting the model with a latent truncation index and a simple reweighting of the atoms, which yields a conditional random finite-atom measure whose marginalized distribution matches the original SSP. This yields at least two consequences: (i) distributionally exact simulation for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the conditional approximation error when this truncation is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.

翻译：离散随机概率测度是贝叶斯推断的核心，尤其作为混合建模与聚类分析中的先验分布。一类广泛且统一的模型是正规物种抽样过程（SSPs），其涵盖了许多贝叶斯非参数先验。我们证明，任何正规SSP均可通过引入潜在截断索引及原子权重的简单重加权，获得精确的条件有限混合表示，从而产生一个条件随机有限原子测度，其边缘化分布与原SSP完全一致。这至少带来两个重要结果：（i）无需用户设定截断水平，即可对任意SSP实现分布精确的模拟；（ii）通过标准有限混合模型机制实现SSP混合模型的后验推断，从而构建无需临时截断的可处理MCMC算法。我们通过推导固定截断条件下条件近似误差的显式全变差界，并重点针对狄利克雷与几何SSPs研究混合建模中的实际性能，深入探讨了这些结果。