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Dirichlet Process mixture models (DPMM) in combination with Gaussian kernelshave been an important modeling tool for numerous data domains arising frombiological, physical, and social sciences. However, this versatility inapplications does not extend to strong theoretical guarantees for theunderlying parameter estimates, for which only a logarithmic rate is achieved.In this work, we (re)introduce and investigate a metric, namedOrlicz-Wasserstein distance, in the study of the Bayesian contraction behaviorfor the parameters. We show that despite the overall slow convergenceguarantees for all the parameters, posterior contraction for parameters happensat almost polynomial rates in outlier regions of the parameter space. Ourtheoretical results provide new insight in understanding the convergencebehavior of parameters arising from various settings of hierarchical Bayesiannonparametric models. In addition, we provide an algorithm to compute themetric by leveraging Sinkhorn divergences and validate our findings through asimulation study.

On Excess Mass Behavior in Gaussian Mixture Models with Orlicz-Wasserstein Distances