{"title":"丰富的皮特曼-你的过程。","authors":"Tommaso Rigon, Sonia Petrone, Bruno Scarpa","doi":"10.1111/sjos.12765","DOIUrl":null,"url":null,"abstract":"<p><p>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.</p>","PeriodicalId":49567,"journal":{"name":"Scandinavian Journal of Statistics","volume":"52 2","pages":"631-657"},"PeriodicalIF":1.0000,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12338310/pdf/","citationCount":"0","resultStr":"{\"title\":\"Enriched Pitman-Yor processes.\",\"authors\":\"Tommaso Rigon, Sonia Petrone, Bruno Scarpa\",\"doi\":\"10.1111/sjos.12765\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>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.</p>\",\"PeriodicalId\":49567,\"journal\":{\"name\":\"Scandinavian Journal of Statistics\",\"volume\":\"52 2\",\"pages\":\"631-657\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12338310/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Scandinavian Journal of Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1111/sjos.12765\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2025/1/19 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Scandinavian Journal of Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1111/sjos.12765","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/1/19 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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.
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.