August 2023 A unified construction for series representations and finite approximations of completely random measures
Juho Lee, Xenia Miscouridou, François Caron
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Bernoulli 29(3): 2142-2166 (August 2023). DOI: 10.3150/22-BEJ1536

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 (generalised) 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 a generalisation of constructions based on size-biased sampling of Poisson point process (Probab. Theory Related Fields 92 (1992) 21–39). It includes as special cases several known series representations and finite approximations as well as novel ones. In particular, we show that one can get novel series representations for the generalised gamma process and the stable beta process. We show how these constructions can be used to derive novel algorithms for posterior inference, including a generalisation of the slice sampler for normalised CRMs mixture models introduced by (J. Comput. Graph. Statist. 20 (2011) 241–259). We also provide some analysis of the truncation error.

Citation

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Juho Lee. Xenia Miscouridou. François Caron. "A unified construction for series representations and finite approximations of completely random measures." Bernoulli 29 (3) 2142 - 2166, August 2023. https://doi.org/10.3150/22-BEJ1536

Information

Received: 1 August 2021; Published: August 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580911
zbMATH: 07691576
Digital Object Identifier: 10.3150/22-BEJ1536

Keywords: Bayesian nonparametrics , completely random measures , generalised gamma process , Poisson random measures , slice sampling

Vol.29 • No. 3 • August 2023
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