[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82352-en":3,"doc-seo-82352-105":29,"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":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":13,"seo_description":14,"update_tm":27,"read_time":28},82352,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","How Does Bayesian Causal Discovery Fail Characterising Structural Consequences in Linear Gaussian Networks under Latent Confounding","Bayesian causal discovery quantifies epistemic uncertainty over DAGs via posterior inference, yet latent confounding has remained insufficiently characterized in terms of posterior behaviour. This work studies linear Gaussian causal models with additive latent confounding between exactly two observed variables, deriving a critical correlation threshold where the score favours a spurious edge. The threshold decreases with sample size, and two distinct posterior failure regimes emerge from the local structure around the confounded variables, validated by exact posterior computations.","arXiv :2607 .09449v 1 [ cs .AI] 10 Jul 2026  \nHow Does Bayesian Causal Discovery Fail? Characterising Structural Consequences in Linear Gaussian Networks under Latent  \nConfounding  \nDebargha Ghosh 1 , Silja Renooij 1 , and Anna Kononova2  \n1 Department of Information and Computing Sciences, Utrecht University  \n2 Leiden Institute of Advanced Computer Science, Leiden University  \nAbstract  \nBayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size – more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.  \n1 Introduction  \nDiscovering the causal relationships underlying a system of variables is a central problem across scientific domains. Bayesian networks and structural equation models provide a principled framework for representing such dependencies [11, 17] . Structure learning methods aim to recover a graph representing the underlying data generating process from observational data, but are inherently limited by statistical uncertainty and identifiability constraints [8] . In contrast, Bayesian causal structure learning infers a posterior distribution p(G | D) over candidate directed acyclic graphs (DAGs) representing the underlying causal system [5], enabling principled quantification of epistemic uncertainty. This distribution allows assessing confidence in inferred structures and supports downstream tasks such as experimental design and active causal discovery [16, 22], where uncertainty over candidate graphs plays a central role. However, how this posterior uncertainty changes under violations of modelling assumptions, such as latent confounding, is not well understood.  \nThe reliability of posterior-based causal conclusions from observational data—namely, whether posterior probabilities faithfully reflect uncertainty over plausible structures—depends on two key components: identifiability and scoring. On the identifiability side, under the assumptions of the causal Markov condition, causal sufficiency (lack of latent confounding), and faithfulness, observational data can distinguish causal structures only up to Markov equivalence [23, 25], even under infinite data. On the scoring side, the choice of score—such as the Bayesian Gaussian equivalent (BGe) score in linear Gaussian models [7, 12]—determines the likelihood and prior that govern how evidence is translated into posterior probabilities over candidate causal structures, under assumptions such as independent exogenous noise and Gaussianity. While both  \nidentifiability and scoring are well understood under standard assumptions, it remains unclear how their interplay affects posterior behaviour when these assumptions are violated.  \nOf these assumptions, causal sufficiency is particularly consequential for causal discovery from observational data. The assumption is not testable in practice, yet it is routinely violated in many applications. Causal discovery methods are widely applied to settings such as gene regulatory network inference from expression data [1, 6, 19] . H","cbCaidLy2MAzwUTl","https://ap.wps.com/l/cbCaidLy2MAzwUTl","pdf",483077,1,13,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What problem does the paper address in Bayesian causal discovery?\",\"answer\":\"It analyzes how Bayesian posterior uncertainty over DAGs behaves when the data generation includes latent confounding, rather than only noting that confounding breaks identifiability.\"},{\"question\":\"How is the latent confounding modelled in this study?\",\"answer\":\"The study focuses on linear Gaussian causal models with additive latent confounding between exactly two observed variables.\"},{\"question\":\"What determines when Bayesian methods favour a spurious edge?\",\"answer\":\"A derived critical correlation threshold governs when the score function prefers graphs containing an edge between the confounded variables, and this threshold decreases as sample size increases.\"}]",1784179828,33,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"how-does-bayesian-causal-discovery-fail-characterising-structural-consequences-in-linear-gaussian-networks-under-latent-confounding","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/how-does-bayesian-causal-discovery-fail-characterising-structural-consequences-in-linear-gaussian-networks-under-latent-confounding/82352/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 Bayesian causal discovery?","Question",{"text":75,"@type":76},"It analyzes how Bayesian posterior uncertainty over DAGs behaves when the data generation includes latent confounding, rather than only noting that confounding breaks identifiability.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the latent confounding modelled in this study?",{"text":80,"@type":76},"The study focuses on linear Gaussian causal models with additive latent confounding between exactly two observed variables.",{"name":82,"@type":73,"acceptedAnswer":83},"What determines when Bayesian methods favour a spurious edge?",{"text":84,"@type":76},"A derived critical correlation threshold governs when the score function prefers graphs containing an edge between the confounded variables, and this threshold decreases as sample size increases.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]