[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-147699-105":59,"doc-detail-147699-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","tree-structured-stick-breaking-for-hierarchical-data-bayesian-nonparametric-prior","Tree-Structured Stick Breaking for Hierarchical Data - Bayesian Nonparametric Prior","","Many data are naturally modeled by an unobserved hierarchical structure, motivating a flexible nonparametric prior over unknown data hierarchies. The method uses nested stick-breaking processes to generate trees with unbounded width and depth, allowing data to reside at any node and ensuring infinite exchangeability. The model can be interpreted as an infinite mixture with a dependency structure shaped by an evolutionary diffusion down the tree. Markov chain Monte Carlo with slice sampling enables Bayesian inference and posterior simulation on trees, applied to hierarchical image clustering and text topic modeling.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/tree-structured-stick-breaking-for-hierarchical-data-bayesian-nonparametric-prior/147699/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/tree-structured-stick-breaking-for-hierarchical-data-bayesian-nonparametric-prior/147699.png","ImageObject",300,407,{"name":92,"@type":93},"Ava Thompson","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-08","2026-08-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What hierarchical structure does the proposed model assume?","Question",{"text":113,"@type":114},"The model assumes an unobserved hierarchical structure over which a prior is defined, forming trees with potentially unbounded width and depth.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does nested stick-breaking produce a tree-structured partition?",{"text":118,"@type":114},"Nested stick-breaking interleaves multiple stick-breaking processes, yielding a hierarchical partition where components can correspond to nodes across the tree.",{"name":120,"@type":111,"acceptedAnswer":121},"How are inference and posterior simulation performed?",{"text":122,"@type":114},"Inference uses Markov chain Monte Carlo methods based on slice sampling, allowing simulation from the posterior distribution on trees.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},147699,1787768523,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":44,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144},1649267921044,"https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1786009248482753345","Tree-Structured Stick Breaking for Hierarchical Data  \nRyan Prescott Adams􀀃  \nDept. of Computer Science University of Toronto  \nZoubin Ghahramani  \nDept. of Engineering University of Cambridge  \nMichael I. Jordan  \nDepts. of EECS and Statistics University of California, Berkeley  \nAbstract  \nMany data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies.  \nThe approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable.  \nOne can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.  \n1 Introduction  \nStructural aspects of models are often critical to obtaining flexible, expressive model families. In many cases, however, the structure is unobserved and must be inferred, either as an end in itself or to assist in other estimation and prediction tasks. This paper addresses an important instance of the structure learning problem: the case when the data arise from a latent hierarchy. We take a direct nonparametric Bayesian approach, constructing a prior on tree-structured partitions of data that provides for unbounded width and depth while still allowing tractable posterior inference.  \nProbabilistic approaches to latent hierarchies have been explored in a variety of domains. Unsupervised learning of densities and nested mixtures has received particular attention via finite-depth trees [1], diffusive branching processes [2] and hierarchical clustering [3, 4] . Bayesian approaches to learning latent hierarchies have also been useful for semi-supervised learning [5], relational learning [6] and multi-task learning [7] . In the vision community, distributions over trees have been useful as priors for figure motion [8] and for discovering visual taxonomies [9] .  \nIn this paper we develop a distribution over probability measures that imbues them with a natural hierarchy. These hierarchies have unbounded width and depth and the data may live at internal nodes on the tree. As the process is defined in terms of a distribution over probability measures and not as a distribution over data per se, data from this model are infinitely exchangeable; the probability of any set of data is not dependent on its ordering. Unlike other infinitely exchangeable models [2, 4], a pseudo-time process is not required to describe the distribution on trees and it can be understood in terms of other popular Bayesian nonparametric models.  \nOur new approach allows the components of an infinite mixture model to be interpreted as part of a diffusive evolutionary process. Such a process captures the natural structure of many data. For example, some scientific papers are considered seminal—they spawn new areas of research and cause new papers to be written. We might expect that within a text corpus of scientific documents, such papers would be the natural ancestors of more specialized papers that followed on from the new ideas. This motivates two desirable features of a distribution over hierarchies: 1) ancestor data (the  \n􀀃 [http://www](http://www) .cs .toronto .edu/˜rpa/  \n(a) Dirichlet process stick breaking  \n(b) Tree-structured stick breaking  \nFigure 1: a) Dirichlet process stick-breaking procedure, with a linear partitioning. b) Interleaving two stickbreaking processes yields a tree-structured partition. Rows 1, 3 and 5 are 􀀗-breaks. Rows 2 and 4 are -breaks.“prototypes”) should be able to live at internal nodes in the tree, and 2) as the ancestor/descendant relationships are not known a priori, the data ","cbCaic3f3tVS01oG","https://ap.wps.com/l/cbCaic3f3tVS01oG","pdf",1238574,"English","# Introduction\n## A distribution over unobserved latent hierarchies\n## Motivation and key properties\n# A Tree-Structured Stick-Breaking Process\n## Dirichlet process stick-breaking background\n## Extending GEM to tree-structured partitions","[{\"question\":\"What hierarchical structure does the proposed model assume?\",\"answer\":\"The model assumes an unobserved hierarchical structure over which a prior is defined, forming trees with potentially unbounded width and depth.\"},{\"question\":\"How does nested stick-breaking produce a tree-structured partition?\",\"answer\":\"Nested stick-breaking interleaves multiple stick-breaking processes, yielding a hierarchical partition where components can correspond to nodes across the tree.\"},{\"question\":\"How are inference and posterior simulation performed?\",\"answer\":\"Inference uses Markov chain Monte Carlo methods based on slice sampling, allowing simulation from the posterior distribution on trees.\"}]","Tree-Structured Stick Breaking for Hierarchical Data - Bayesian Nonparametric Prior | PDF",23]