[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84191-en":3,"doc-seo-84191-105":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":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},84191,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Recovering Latent Structures after Variational Bayesian Variable Selection","Partially exploratory factor analysis (PEFA) only weakly specifies the loading structure and factor count, while PCFA-VA recovers structure via Bayesian variable selection using spike-and-slab priors and variational inference. The work develops post-selection fit assessment for hard- and soft-selected covariance models, deriving parameter counts, degrees of freedom, absolute diagnostics (RMSEA, SRMR, CFI, TLI) and relative criteria (AIC, BIC, ELBO). A scale-free gain rule selects factors and is proven to recover bracketed true dimensionality, supported by simulations and a PID-5 application.","arXiv :2607 .07159v1 [ stat .ME] 8 Jul 2026  \nRecovering Latent Structures after Variational Bayesian Variable Selection: Fit Assessment and Factor-Number Selection in Partially  \nExploratory Factor Analysis  \nJinsong Chena 1 and Yi Jinb  \naFaculty of Education  \nThe University of Hong Kong  \nbDepartment of Curriculum and Instruction  \nThe Education University of Hong Kong  \nIn partially exploratory factor analysis (PEFA), the loading structure and the number of factors are only weakly specified, and the regularized variational approximation for partially confirmatory factor analysis (PCFA-VA) recovers the structure by Bayesian variable selection: spikeand-slab priors with variational inference attach an inclusion probability to every unspecified loading. This research develops the post-selection assessment that such variable selection requires. We convert the converged solution into a covariance model under hard selection, which thresholds the inclusion probabilities into a fixed sparse pattern, or soft selection, which retains them as weights and yields an effective number of parameters in the spirit of Bayesian and generalized degrees of freedom. We derive the corresponding parameter counts, degrees of freedom, absolute fit diagnostics (RMSEA, SRMR, CFI, TLI), and relative criteria (AIC, BIC, and the algorithm-native evidence lower bound, ELBO) . To choose the number of factors we propose a scale-free gain rule with a sustained-drop guard, which retains a factor only when its marginal gain in a criterion exceeds a fixed fraction of the largest gain, and we give conditions under which it exactly recovers a bracketed true dimensionality. Two simulation studies show that the absolute indices track loading recovery and flag under-factoring, that the raw criteria over-factor whereas the gain rule recovers the true dimensionality across a wide range of its threshold, and that the ELBO gain is the most robust variant under heavy nuisance. An empirical example with the 100-item PID-5 shows that the selected solution fits better than the confirmatory 25-facet model even under a plug-in evaluation that disfavors it, that two fully disjoint specifications recover the major latent structure concordantly, and that the specification  \ncontrols the resolution of mutually collinear factors.  \nKeywords: Bayesian variable selection; spike-and-slab priors; partially exploratory factor  \nanalysis; variational approximation; model fit; effective number of parameters  \nIntroduction  \nPartially confirmatory factor analysis (PCFA) was proposed to bridge the practical gap between exploratory factor analysis (EFA), where little prior knowledge is imposed, and confirmatory factor analysis (CFA), where the loading matrix is strongly specified (Chen et al., 2021) . The regularized variational approximation for PCFA (PCFA-VA) further improves this framework by replacing computationally intensive MCMC estimation with a deterministic variational optimization scheme and by using stochastic search variable selection (SSVS) to identify active and inactive unspecified loadings (Jin & Chen, 2025) . In PCFA-VA, entries of the specification matrix Q may be required, unrequired, or unspecified, allowing different amounts of substantive knowledge to be incorporated in the same model. Recovering a partially specified loading structure is thus, at its core, a  \nBayesian variable-selection problem in a high-dimensional latent variable model: each unspecified loading is a candidate variable, the spike-and-slab prior supplies the selection mechanism, and the converged variational posterior attaches an inclusion probability to every candidate.  \nIn this research, we expand the PCFA-VA machinery to accommodate partially exploratory factor analysis (PEFA), where the loading structure and, potentially, the number offactors are partially, and even weakly, specified. The algorithm, the notation, the priors, the variational posterior moments, and the loading activeness rules ","cbCaifZyOUtIlWVw","https://ap.wps.com/l/cbCaifZyOUtIlWVw","pdf",879337,4,1,23,"English","en",105,"# Introduction\n## Analytical Framework\n## The Logic of PEFA with Fit Statistics","[{\"question\":\"What problem does the paper address in partially exploratory factor analysis (PEFA)?\",\"answer\":\"PEFA weakly specifies the loading structure and may leave the number of factors uncertain. The paper addresses how to assess fit after applying variational Bayesian variable selection.\"},{\"question\":\"How does the method recover latent loading structure in PCFA-VA?\",\"answer\":\"It uses spike-and-slab priors with variational inference, assigning an inclusion probability to each unspecified loading as part of Bayesian variable selection.\"},{\"question\":\"What is proposed for selecting the number of factors in PEFA?\",\"answer\":\"A scale-free gain rule with a sustained-drop guard keeps a factor only when its marginal gain exceeds a fixed fraction of the largest gain, with conditions under which it recovers the correct bracketed dimensionality.\"}]",1784193837,58,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"recovering-latent-structures-after-variational-bayesian-variable-selection","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/recovering-latent-structures-after-variational-bayesian-variable-selection/84191/",{"url":52,"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","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in partially exploratory factor analysis (PEFA)?","Question",{"text":75,"@type":76},"PEFA weakly specifies the loading structure and may leave the number of factors uncertain. The paper addresses how to assess fit after applying variational Bayesian variable selection.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method recover latent loading structure in PCFA-VA?",{"text":80,"@type":76},"It uses spike-and-slab priors with variational inference, assigning an inclusion probability to each unspecified loading as part of Bayesian variable selection.",{"name":82,"@type":73,"acceptedAnswer":83},"What is proposed for selecting the number of factors in PEFA?",{"text":84,"@type":76},"A scale-free gain rule with a sustained-drop guard keeps a factor only when its marginal gain exceeds a fixed fraction of the largest gain, with conditions under which it recovers the correct bracketed dimensionality.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"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":20,"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"]