Enriched Pitman-Yor processes.

IF 1 4区 数学 Q3 STATISTICS & PROBABILITY
Scandinavian Journal of Statistics Pub Date : 2025-06-01 Epub Date: 2025-01-19 DOI:10.1111/sjos.12765
Tommaso Rigon, Sonia Petrone, Bruno Scarpa
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引用次数: 0

Abstract

Bayesian nonparametrics has evolved into a broad area encompassing flexible methods for Bayesian inference, combinatorial structures, tools for complex data reduction, and more. Discrete prior laws play an important role in these developments, and various choices are available nowadays. However, many existing priors, such as the Dirichlet process, have limitations if data require nested clustering structures. Thus, we introduce a discrete nonparametric prior, termed the enriched Pitman-Yor process, which offers higher flexibility in modeling such elaborate partition structures. We investigate the theoretical properties of this novel prior and establish its formal connection with the enriched Dirichlet process and normalized random measures. Additionally, we present a square-breaking representation and derive closed-form expressions for the posterior law and associated urn schemes. Furthermore, we demonstrate that several established models, including Dirichlet processes with a spike-and-slab base measure and mixture of mixtures models, emerge as special instances of the enriched Pitman-Yor process, which therefore serves as a unified probabilistic framework for various Bayesian nonparametric priors. To illustrate its practical utility, we employ the enriched Pitman-Yor process for a species-sampling ecological problem.

丰富的皮特曼-你的过程。
贝叶斯非参数已经发展成为一个广泛的领域,包括贝叶斯推理的灵活方法、组合结构、复杂数据简化的工具等等。离散先验律在这些发展中起着重要作用,目前有多种选择。然而,许多现有的先验,如Dirichlet过程,在数据需要嵌套聚类结构时具有局限性。因此,我们引入了一个离散的非参数先验,称为丰富的Pitman-Yor过程,它在建模这种复杂的分区结构时提供了更高的灵活性。我们研究了这种新先验的理论性质,并建立了它与富狄利克雷过程和归一化随机测度的形式联系。此外,我们给出了后验律的破方表示,并推导出了后验律和相关的瓮形方案的封闭表达式。此外,我们还证明了几个已建立的模型,包括具有尖峰-板基础测量的Dirichlet过程和混合模型,作为丰富的Pitman-Yor过程的特殊实例,因此可以作为各种贝叶斯非参数先验的统一概率框架。为了说明其实际用途,我们采用了丰富的皮特曼-尤尔过程物种采样的生态问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Scandinavian Journal of Statistics
Scandinavian Journal of Statistics 数学-统计学与概率论
CiteScore
1.80
自引率
0.00%
发文量
61
审稿时长
6-12 weeks
期刊介绍: The Scandinavian Journal of Statistics is internationally recognised as one of the leading statistical journals in the world. It was founded in 1974 by four Scandinavian statistical societies. Today more than eighty per cent of the manuscripts are submitted from outside Scandinavia. It is an international journal devoted to reporting significant and innovative original contributions to statistical methodology, both theory and applications. The journal specializes in statistical modelling showing particular appreciation of the underlying substantive research problems. The emergence of specialized methods for analysing longitudinal and spatial data is just one example of an area of important methodological development in which the Scandinavian Journal of Statistics has a particular niche.
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