Hierarchical Normalized Completely Random Measures to Cluster Grouped Data

Abstract

In this article, we propose a Bayesian nonparametric model for clustering grouped data. We adopt a hierarchical approach: at the highest level, each group of data is modeled according to a mixture, where the mixing distributions are conditionally independent normalized completely random measures (NormCRMs) centered on the same base measure, which is itself a NormCRM. The discreteness of the shared base measure implies that the processes at the data level share the same atoms. This desired feature allows to cluster together observations of different groups. We obtain a representation of the hierarchical clustering model by marginalizing with respect to the infinite dimensional NormCRMs. We investigate the properties of the clustering structure induced by the proposed model and provide theoretical results concerning the distribution of the number of clusters, within and between groups. Furthermore, we offer an interpretation in terms of generalized Chinese restaurant franchise process, which allows for posterior inference under both conjugate and nonconjugate models. We develop algorithms for fully Bayesian inference and assess performances by means of a simulation study and a real-data illustration. Supplementary materials for this article are available online.

Keywords:

• Bayesian nonparametrics
• Clustering
• Hierarchical models
• Mixture models

Acknowledgment

Andrea Cremaschi thanks the Norway Centre for Molecular Medicine (NCMM) IT facility for the computational support.

Additional information

Funding

Raffaele Argiento gratefully acknowledges Collegio Carlo Alberto for partially funding this work.