Criterion for the resemblance between the mother and the model distribution

Criterion for the resemblance between the mother and the model distribution | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Criterion for the resemblance between the mother and the model distribution Yo Sheena This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3105959/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract If the probability distribution model aims to approximate the hidden mother distribution, it is imperative to establish a useful criterion for the resemblance between the mother and the model distributions. This study proposes a criterion that measures the Hellinger distance between discretized (quantized) samples from both distributions. Unlike information criteria such as AIC, this criterion does not require the probability density function of the model distribution, which cannot be explicitly obtained for a complicated model such as a deep learning machine. Second, it can draw a positive conclusion (i.e., both distributions are sufficiently close) under a given threshold, whereas a statistical hypothesis test, such as the Kolmogorov-Smirnov test, cannot genuinely lead to a positive conclusion when the hypothesis is accepted. In this study, we establish a reasonable threshold for the criterion deduced from the Bayes error rate and also present the asymptotic bias of the estimator of the criterion. From these results, a reasonable and easy-to-use criterion is established that can be directly calculated from the two sets of samples from both distributions. MSC(2010) Subject Classification: Primary 60F99, Secondary 62F12 Hellinger distance Kolmogorov-Smirnov test asymptotic risk f-divergence alpha-divergence Bayes error rate Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3105959","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":213323121,"identity":"72c3b718-a85a-4311-9c23-fb61ae01eb8d","order_by":0,"name":"Yo Sheena","email":"data:image/png;base64,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","orcid":"","institution":"Shiga University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yo","middleName":"","lastName":"Sheena","suffix":""}],"badges":[],"createdAt":"2023-06-25 07:29:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3105959/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3105959/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":39221266,"identity":"6b137341-db9a-4c16-93fc-5beaadde43c9","added_by":"auto","created_at":"2023-06-28 09:14:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":342767,"visible":true,"origin":"","legend":"","description":"","filename":"submission.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3105959/v1_covered_b0fd2b28-07cc-4669-abf9-0ced5b78a389.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Criterion for the resemblance between the mother and the model distribution","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Hellinger distance, Kolmogorov-Smirnov test, asymptotic risk, f-divergence, alpha-divergence, Bayes error rate","lastPublishedDoi":"10.21203/rs.3.rs-3105959/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3105959/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIf the probability distribution model aims to approximate the hidden mother distribution, it is imperative to establish a useful criterion for the resemblance between the mother and the model distributions.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThis study proposes a criterion that measures the Hellinger distance between discretized (quantized) samples from both distributions. 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