  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.
