The nonparametric view of Bayesian inference has transformed statistics and many of its applications. The canonical Dirichlet process and other more general families of nonparametric priors have served as a gateway to solve frontier uncertainty quantification problems of large, or infinite, nature. This success has been greatly due to available constructions and representations of such distributions, which in turn have lead to a variety of sampling schemes. Undoubtedly, the two most useful constructions are the one based on normalization of homogeneous completely random measures and that based on stick-breaking processes, as well as various particular cases. Understanding their distributional features and how different random probability measures compare among themselves is a key ingredient for their proper application. In this paper, we explore the prior discrepancy, through a divergence-based analysis, of extreme classes of stick-breaking processes. Specifically, we investigate the random Kullback-Leibler divergences between the Dirichlet process and the geometric process, as well as some of their moments. Furthermore, we also perform the analysis within the general exchangeable stick-breaking class of nonparametric priors, leading to appealing results.

翻译：贝叶斯推断的非参数视角已经彻底改变了统计学及其众多应用领域。经典的狄利克雷过程以及其他更一般的非参数先验族已成为解决大规模或无限性质前沿不确定性量化问题的关键工具。这一成功很大程度上归功于这类分布的有效构造和表示方法，进而催生了多种采样方案。无疑，最有用的两种构造是基于齐次完全随机测度归一化的方法和基于棍棒断裂过程的方法，以及它们的各种特例。理解这些分布的特征以及不同随机概率度量之间的比较方式，是正确应用它们的关键要素。本文通过基于散度的分析，探讨了棍棒断裂过程极端类别的先验差异。具体而言，我们研究了狄利克雷过程与几何过程之间的随机Kullback-Leibler散度及其部分矩。此外，我们还在一般可交换棍棒断裂非参数先验类中进行了分析，得出了有趣的结果。