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, the two most useful constructions are the one based on normalization of homogeneous completely random measures and that based on stick-breaking processes. Hence, 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 analyse the discrepancy among some nonparametric priors employed in the literature. Initially, we compute the mean and variance of the random Kullback-Leibler divergence between the Dirichlet process and the geometric process. Subsequently, we extend our analysis to encompass a broader class of exchangeable stick-breaking processes, which includes the Dirichlet and geometric processes as extreme cases. Our results establish quantitative conditions where all the aforementioned priors are close in total variation distance. In such instances, adhering to Occam's razor principle advocates for the preference of the simpler process.

翻译：贝叶斯推断的非参数观点已经彻底改变了统计学及其众多应用。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的前沿不确定性量化问题的关键工具。这一成功在很大程度上归功于这些分布的可实现构造与表示，其中两种最有用的构造是基于齐次完全随机测度归一化的方法以及基于棍棒断裂过程的方法。因此，理解其分布特征以及不同随机概率度量之间的比较方法，是其恰当应用的关键要素。本文分析了文献中使用的若干非参数先验之间的差异。首先，我们计算了狄利克雷过程与几何过程之间的随机Kullback-Leibler散度的均值和方差。随后，我们将分析扩展到一类更广泛的可交换棍棒断裂过程，该过程包含狄利克雷过程和几何过程作为极端情形。我们的结果建立了上述所有先验在总变差距离上相近的定量条件。在此类情形下，遵循奥卡姆剃刀原则主张优先选择更简单的过程。