Exact finite mixture representations for species sampling processes

Random probability measures, together with their constructions, representations, and associated algorithms, play a central role in modern Bayesian inference. A key class is that of proper species sampling processes, which offer a relatively simple yet versatile framework that extends naturally to non-exchangeable settings. We revisit this class from a computational perspective and show that they admit exact finite mixture representations. In particular, we prove that any proper species sampling process can be written, at the prior level, as a finite mixture with a latent truncation variable and reweighted atoms, while preserving its distributional features exactly. These finite formulations can be used as drop-in replacements in Bayesian mixture models, recasting posterior computation in terms of familiar finite-mixture machinery. This yields straightforward MCMC implementations and tractable expressions, while avoiding ad hoc truncations and model-specific constructions. The resulting representation preserves the full generality of the original infinite-dimensional priors while enabling practical gains in algorithm design and implementation.