A unified construction for series representations and finite approximations of completely random measures
This work addresses a methodological bottleneck for researchers in Bayesian nonparametrics, offering incremental improvements in simulation and inference techniques for complex models.
The authors tackled the challenge of exact simulation and scalable inference with infinite-activity completely random measures (CRMs) by proposing a unified framework for deriving series representations and finite-dimensional approximations, which includes novel representations for the generalized gamma process and stable beta process and provides truncation error analysis.
Infinite-activity completely random measures (CRMs) have become important building blocks of complex Bayesian nonparametric models. They have been successfully used in various applications such as clustering, density estimation, 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 simulation or scalable inference with these models is challenging and finite-dimensional approximations are often considered. In this work, we propose a general and unified framework to derive both series representations and finite-dimensional approximations of CRMs. Our framework can be seen as an extension of constructions based on size-biased sampling of Poisson point process [Perman1992]. It includes as special cases several known series representations as well as novel ones. In particular, we show that one can get novel series representations for the generalized gamma process and the stable beta process. We also provide some analysis of the truncation error.