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 two-stage finite-mixture representation built from a latent truncation index and a simple reweighting of the atoms. For each realized truncation index, the representation has finitely many atoms, and averaging over the induced law of that index recovers the original SSP setwise. This yields at least two consequences: (i) an exact two-stage finite construction 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 approximation error when the truncation level is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.

翻译：离散随机概率测度在贝叶斯推断中核心重要，尤其作为混合建模和聚类的先验。一个广泛且统一的类别是适当的物种抽样过程（SSPs），涵盖了许多贝叶斯非参数先验。我们证明任何适当的SSP都允许一种精确的两阶段有限混合表示，该表示基于潜在截断索引和对原子进行简单重加权构建。对于每个实现的截断索引，该表示具有有限多个原子，并且对该索引诱导律进行平均即可恢复原始SSP的集合性质。这至少产生两个结果：（i）任意SSP的精确两阶段有限构造，无需用户选择的截断水平；（ii）通过标准有限混合机制进行SSP混合模型的后验推断，从而产生无临时截断的可处理MCMC算法。我们通过推导截断水平固定时近似误差的显式全变分界限，并研究混合建模中的实际性能（重点考察狄利克雷和几何SSP），来探索这些结果。