Posterior Distribution for the Number of Clusters in Dirichlet Process Mixture Models
This work addresses a foundational issue in Bayesian nonparametrics for statisticians and machine learning practitioners, offering incremental theoretical insights into DPMM behavior.
The authors tackled the problem of inconsistency in inferring the true number of clusters in Dirichlet process mixture models (DPMM) by rigorously studying the posterior distribution under various priors and data constraints, providing novel lower bounds on probability ratios for cluster counts with Gaussian or uniform priors.
Dirichlet process mixture models (DPMM) play a central role in Bayesian nonparametrics, with applications throughout statistics and machine learning. DPMMs are generally used in clustering problems where the number of clusters is not known in advance, and the posterior distribution is treated as providing inference for this number. Recently, however, it has been shown that the DPMM is inconsistent in inferring the true number of components in certain cases. This is an asymptotic result, and it would be desirable to understand whether it holds with finite samples, and to more fully understand the full posterior. In this work, we provide a rigorous study for the posterior distribution of the number of clusters in DPMM under different prior distributions on the parameters and constraints on the distributions of the data. We provide novel lower bounds on the ratios of probabilities between $s+1$ clusters and $s$ clusters when the prior distributions on parameters are chosen to be Gaussian or uniform distributions.