[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121656-en":3,"doc-seo-121656-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121656,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Bayesian Renormalization - Bayesian Renormalization","Bayesian Renormalization develops a fully information-theoretic formulation of renormalization inspired by Bayesian statistical inference. It identifies the Fisher metric as defining a correlation length that acts as an emergent RG scale, measuring distinguishability between nearby probability distributions. This scale bounds the achievable precision of an effective model. When applied to physical systems, the emergent scale matches the maximum experimentally probeable energy, recovering ordinary renormalization, while remaining broadly applicable to data science settings such as compression and generative modeling.","Bayesian Renormalization  \narXiv :2305 . 10491v2 [hep-th] 27 May 2023  \nDavid S. Berman 1 , Marc S. Klinger2 , Alexander G. Stapleton 1  \n1 Centre for Theoretical Physics, Queen Mary University of London, Mile End Road, London E1 4NS  \n2 Department of Physics, University of Illinois, Urbana IL 61801, USA  \nAbstract  \nIn this note we present a fully information theoretic approach to renormalization inspired by Bayesian statistical inference, which we refer to as Bayesian Renormalization. The main insight of Bayesian Renormalization is that the Fisher metric de􀀌nes a correlation length that plays the role of an emergent RG scale quantifying the distinguishability between nearby points in the space of probability distributions. This RG scale can be interpreted as a proxy for the maximum number of unique observations that can be made about a given system during a statistical inference experiment. The role of the Bayesian Renormalization scheme is subsequently to prepare an e􀀋ective model for a given system up to a precision which is bounded by the aforementioned scale. In applications of Bayesian Renormalization to physical systems, the emergent information theoretic scale is naturally identi􀀌ed with the maximum energy that can be probed by current experimental apparatus, and thus Bayesian Renormalization coincides with ordinary renormalization. However, Bayesian Renormalization is su􀀎ciently general to apply even in circumstances in which an immediate physical scale is absent, and thus provides an ideal approach torenormalization in data science contexts. To this end, we provide insight into how the Bayesian Renormalization scheme relates to existing methods for data compression and data generation such as the information bottleneck and the di􀀋usion learning paradigm.  \nContents  \n1 Introduction 2  \n2 Renormalization and Di􀀋usion 4  \n2.1 Exact Renormalization is Di􀀋usion ................................ 5  \n2.2 Di􀀋usion is Exact Renormalization ................................ 7  \n3 Bayesian Renormalization and Information Geometry 9  \n3.1 Bayesian Inference and Information Geometry .......................... 9  \n3.2 Dynamical Bayesian Inference ................................... 11  \n3.3 Backward Inference and Model Space Renormalization ..................... 12  \n3.4 Bayesian Inversion and Data Space Renormalization ...................... 13  \n4 Discussion 15  \n1 Introduction  \nIn [1] the question was posed: How does our understanding of a system improve as we obtain more data? The natural language for formulating this question is through statistical inference. From the perspective of statistical inference, our understanding of a system is encoded in the probability we assign to di􀀋erent plausible explanations for how the system works. These explanations are formalized as probability models for observable data speci􀀌ed in terms of various parameters. The probability assigned to each of these models is subsequently encoded in an object called the Bayesian posterior distribution, which can bethought of as a probability distribution over all possible probability distributions for observable data. In terms of these concepts, we successfully formulated an answer to the aforementioned question by deriving an explicit equation governing the evolution of the posterior distribution as a function of the amount of collected data. We refer to this equation and more broadly to the idea of dynamically updating one’s beliefs using Bayesian inference as Dynamical Bayesian Inference, or Dynamical Bayes (DB) . A central observation from DB is that as new data is collected the “current” most likely model 􀀍ows through the space of possible models towards the probability distribution truly responsible for generating observed data.  \nThe idea that learning induces a 􀀍ow in the space of models is immediately quite evocative of adi􀀋erent kind of “meta”-theory: the Renormalization Group (RG) . RG is a set of ideas and strategies broadly concerned with","cbCaih9GG8T9vwSx","https://ap.wps.com/l/cbCaih9GG8T9vwSx","pdf",290361,1,21,"English","en",105,"# Introduction\n## How learning motivates dynamical Bayes\n## Renormalization group as scale in physical theories\n# Renormalization and Diffusion\n## Exact renormalization is diffusion\n## Diffusion is exact renormalization\n# Bayesian Renormalization and Information Geometry\n## Bayesian inference and information geometry\n## Dynamical Bayesian inference\n## Backward inference and model space renormalization\n## Bayesian inversion and data space renormalization\n# Discussion","[{\"question\":\"What is Bayesian Renormalization in this work?\",\"answer\":\"It is an information-theoretic approach to renormalization inspired by Bayesian statistical inference, defining an emergent RG notion in terms of distinguishability of probability distributions.\"},{\"question\":\"How does the Fisher metric enter the Bayesian Renormalization framework?\",\"answer\":\"The Fisher metric defines a correlation length that functions as an emergent RG scale, quantifying how distinguishable nearby points are in probability distribution space.\"},{\"question\":\"Why does Bayesian Renormalization reduce to ordinary renormalization in physical applications?\",\"answer\":\"In those cases, the emergent information-theoretic scale aligns with the maximum energy that current experiments can probe, so the scheme coincides with standard renormalization.\"}]","Bayesian Renormalization - Bayesian Renormalization | PDF",1785805987,53,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"bayesian-renormalization-bayesian-renormalization","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/bayesian-renormalization-bayesian-renormalization/121656/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is Bayesian Renormalization in this work?","Question",{"text":75,"@type":76},"It is an information-theoretic approach to renormalization inspired by Bayesian statistical inference, defining an emergent RG notion in terms of distinguishability of probability distributions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the Fisher metric enter the Bayesian Renormalization framework?",{"text":80,"@type":76},"The Fisher metric defines a correlation length that functions as an emergent RG scale, quantifying how distinguishable nearby points are in probability distribution space.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does Bayesian Renormalization reduce to ordinary renormalization in physical applications?",{"text":84,"@type":76},"In those cases, the emergent information-theoretic scale aligns with the maximum energy that current experiments can probe, so the scheme coincides with standard renormalization.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]