[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85634-en":3,"doc-seo-85634-105":30,"detail-sidebar-cat-0-en-105":83},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85634,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Betting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing","A fundamental assumption in statistics and machine learning is that “the future looks like the past,” formalized as exchangeability: the joint data distribution is invariant under permutations. In practice, distribution shifts and concept drift often break this assumption, degrading model performance unless violations are detected early. Conformal test martingales provide a distribution-free sequential testing framework with guaranteed false-alarm control by betting against the uniformity of conformal p-values. The work proposes shifted Legendre polynomial betting rules and a Variational mean-field approximation to enable scalable real-time monitoring of higher-order moment shifts.","arXiv :2606 .20859v2 [ stat .ML] 12 Jul 2026  \nBetting on Moments: Legendre Jumper Martingales for Online Exchangeability Testing  \nJohan Hallberg Szabadv´ary  \nDepartment of Mathematics, Stockholm University [johan. hallberg. szabadvary@math. su. se](johan. hallberg. szabadvary@math. su. se)  \nMay 2026  \nAbstract  \nA fundamental assumption in statistics and machine learning is that“the future looks like the past,” formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.  \nKeywords: Conformal test martingales, Concept drift detection, Sequential exchangeability testing, Shifted Legendre polynomials, Variational mean-field approximation  \n1 Introduction  \nA standard assumption in statistics and machine learning (ML) is that the underlying data-generating distribution remains stable over time, that is, “the fu-  \nture looks like the past.” This can be formalised as the exchangeability assumption, which posits that the joint distribution of the data sequence is invariant under permutations. Exchangeability is more general than the IID assumption, in which data are generated independently from the same probability distribution, but the two are closely related and even equivalent under some conditions (see below) . Violations of exchangeability—due to distribution shifts, concept drift, or other non-stationarities—can compromise the validity and predictive performance of ML models, which motivates the need for reliable online testing procedures that detect such violations promptly while controlling false-alarm rates.  \nClassical fixed-sample tests are ill-suited for this sequential setting because they either inflate type I error when applied repeatedly or require strong parametric assumptions. Conformal test martingales (CTMs) (Vovk, Alexander Gammerman, and Shafer, 2022) provide a flexible, distribution-free framework for sequentially testing exchangeability by monitoring p-values derived from conformal prediction methods. Under exchangeability, these p-values are independent and uniformly distributed on [0 , 1] . Intuitively, a CTM describes the capital process of a player who gambles against the hypothesis that the p-values p 1 , p2 , . . . are uniformly and independently distributed (which they are under exchangeability) such that the betting game is fair (ensured by the martingale property) and that the gambler never risks bankruptcy (see Section 2 for more details) . If the p-values are independent and uniformly distributed, the gambler is not likely to gain much capital, but if they deviate from uniformity or exhibit dependency, she could ","cbCaiprZ1pSPfbj5","https://ap.wps.com/l/cbCaiprZ1pSPfbj5","pdf",1332191,2,1,30,"English","en",105,"# Introduction\n## Exchangeability and IID connection\n## Sequential testing motivation\n# Conformal Test Martingales and martingale-based false-alarm control\n## Conformal p-values under exchangeability\n## Ville’s inequality and betting interpretation\n# Proposed Legendre Jumper Martingales\n## Simple and Product Legendre Jumpers\n## Variational Legendre Jumper and mean-field approximation","[{\"question\":\"How does the Variational Legendre Jumper address the computational cost of the Product Legendre Jumper?\",\"answer\":\"The Product Legendre Jumper incurs exponential growth in the state space (the “jumping tax”). The Variational Legendre Jumper applies a mean-field approximation to reduce complexity to constant time per step with minimal power loss.\"}]",1784205152,76,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"betting-on-moments-legendre-jumper-martingales-for-online-exchangeability-testing","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/betting-on-moments-legendre-jumper-martingales-for-online-exchangeability-testing/85634/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How does the Variational Legendre Jumper address the computational cost of the Product Legendre Jumper?","Question",{"text":75,"@type":76},"The Product Legendre Jumper incurs exponential growth in the state space (the “jumping tax”). The Variational Legendre Jumper applies a mean-field approximation to reduce complexity to constant time per step with minimal power loss.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,114,119,122,126],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":22,"slug":113},"research-report",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]