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Infinite-activity completely random measures (CRMs) have become importantbuilding blocks of complex Bayesian nonparametric models. They have beensuccessfully used in various applications such as clustering, densityestimation, latent feature models, survival analysis or network science.Popular infinite-activity CRMs include the (generalized) gamma process and the(stable) beta process. However, except in some specific cases, exact simulationor scalable inference with these models is challenging and finite-dimensionalapproximations are often considered. In this work, we propose a general andunified framework to derive both series representations and finite-dimensionalapproximations of CRMs. Our framework can be seen as an extension ofconstructions based on size-biased sampling of Poisson point process[Perman1992]. It includes as special cases several known series representationsas well as novel ones. In particular, we show that one can get novel seriesrepresentations for the generalized gamma process and the stable beta process.We also provide some analysis of the truncation error.

A unified construction for series representations and finite approximations of completely random measures