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.

翻译：狄利克雷过程混合模型（DPMM）结合高斯核已成为生物、物理和社会科学等众多数据领域的重要建模工具。然而，这种应用广泛性并未延伸至底层参数估计的强理论保证，目前仅能实现对数收敛速率。本文（重新）引入并研究了一种名为Orlicz-Wasserstein距离的度量，用于探究参数的贝叶斯收缩行为。研究表明，尽管所有参数的整体收敛速度较慢，但在参数空间的异常值区域，后验收缩能以近乎多项式的速率实现。我们的理论结果为理解分层贝叶斯非参数模型不同设定下参数的收敛行为提供了新见解。此外，我们利用Sinkhorn散度设计了该度量的计算算法，并通过仿真研究验证了理论发现。