  Dirichlet Process mixture models (DPMM) in combination with Gaussian kernels
have been an important modeling tool for numerous data domains arising from
biological, physical, and social sciences. However, this versatility in
applications does not extend to strong theoretical guarantees for the
underlying parameter estimates, for which only a logarithmic rate is achieved.
In this work, we (re)introduce and investigate a metric, named
Orlicz-Wasserstein distance, in the study of the Bayesian contraction behavior
for the parameters. We show that despite the overall slow convergence
guarantees for all the parameters, posterior contraction for parameters happens
at almost polynomial rates in outlier regions of the parameter space. Our
theoretical results provide new insight in understanding the convergence
behavior of parameters arising from various settings of hierarchical Bayesian
nonparametric models. In addition, we provide an algorithm to compute the
metric by leveraging Sinkhorn divergences and validate our findings through a
simulation study.
