DISTRIBUTION THEORY FOR HIERARCHICAL PROCESSES

• 成果类型: Article
• 署名作者: Camerlenghi, Federico; Lijoi, Antonio; Orbanz, Peter; Prunster, Igor
• 署名单位: University of Milano-Bicocca; Bocconi University; Bocconi University; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1678
• 发表日期: 2019
• 页码: 67-92
• 关键词: inference; calculus; priors

摘要

Hierarchies of discrete probability measures are remarkably popular as nonparametric priors in applications, arguably due to two key properties: (i) they naturally represent multiple heterogeneous populations; (ii) they produce ties across populations, resulting in a shrinkage property often described as sharing of information. In this paper, we establish a distribution theory for hierarchical random measures that are generated via normalization, thus encompassing both the hierarchical Dirichlet and hierarchical Pitman-Yor processes. These results provide a probabilistic characterization of the induced (partially exchangeable) partition structure, including the distribution and the asymptotics of the number of partition sets, and a complete posterior characterization. They are obtained by representing hierarchical processes in terms of completely random measures, and by applying a novel technique for deriving the associated distributions. Moreover, they also serve as building blocks for new simulation algorithms, and we derive marginal and conditional algorithms for Bayesian inference.