Bayesian mixture models (in)consistency for the number of clusters

Abstract

Bayesian nonparametric mixture models are common for modeling complex data. While these models are well-suited for density estimation, recent results proved posterior inconsistency of the number of clusters when the true number of components is finite, for the Dirichlet process and Pitman--Yor process mixture models. We extend these results to additional Bayesian nonparametric priors such as Gibbs-type processes and finite-dimensional representations thereof. The latter include the Dirichlet multinomial process, the recently proposed Pitman-Yor, and normalized generalized gamma multinomial processes. We show that mixture models based on these processes are also inconsistent in the number of clusters and discuss possible solutions. Notably, we show that a post-processing algorithm introduced for the Dirichlet process can be extended to more general models and provides a consistent method to estimate the number of components.

Added a real-data analysis (Section 6) Revised version (changes in Section 4.1)

References in corpus (11)

• Model-based Clustering
• Sparse graphs using exchangeable random measures
• Model-based clustering based on sparse finite Gaussian mixtures
• Are Gibbs-type priors the most natural generalization of the Dirichlet process?
• Posterior convergence rates of Dirichlet mixtures at smooth densities
• Convergence of latent mixing measures in finite and infinite mixture models
• Bayesian semiparametric inference for multivariate doubly-interval-censored data
• Inconsistency of Pitman-Yor process mixtures for the number of components
• Generalized mixtures of finite mixtures and telescoping sampling
• Bounds for Bayesian order identification with application to mixtures
• Spying on the prior of the number of data clusters and the partition distribution in Bayesian cluster analysis