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

## TL;DR

This paper introduces a unified framework for deriving series representations and finite approximations of completely random measures, enhancing scalability and simulation in Bayesian nonparametrics.

## Contribution

It extends existing constructions to include new series representations for important CRMs like the generalized gamma and stable beta processes.

## Key findings

- Includes known and novel series representations for CRMs.
- Provides analysis of truncation errors in approximations.
- Enables scalable inference in complex Bayesian models.

## Abstract

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.