[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84428-en":3,"doc-seo-84428-105":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},84428,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Meta-Dependence in Conditional Independence Testing","Conditional independence testing underpins feature screening, invariant statistical modeling, and causal discovery, yet many methods rely on sequential CI tests whose stability depends on how test outcomes interact. The work introduces a geometric view: enforcing CI properties restricts possible joint distributions to manifolds, so correlations across CI tests reflect the distribution’s position relative to these manifolds. A computational moment-projection measure is developed, with closed forms for multivariate Gaussians and validation on synthetic and real data, enabling applications such as tuning significance thresholds to improve causal discovery.","Meta-Dependence in Conditional Independence Testing  \nBijan H. S. Mazaheri 1,2 Jiaqi Zhang2, 3 Caroline Uhler2, 3  \n1Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, USA  \n2 Schmidt Center, Broad Institute of MIT and Harvard, Cambridge, Massachusetts, USA  \n3Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA  \narXiv :2504 . 12594v2 [ cs .LG] 13 Jul 2026  \nAbstract  \nConditional independence testing is a critical component of feature screening, invariant statistical models, and causal discovery. Many of these algorithms rely on the sequential application of conditional independence tests, and their stability hingeson how their outcomes interact. We study this“meta-dependence” between conditional independence properties using the following geometric intuition: satisfying each conditional independence property constrains the space of possible joint distributions to a manifold. The “meta-dependence”  \nof multiple conditional independences in a probability distribution is informed by its position relative to these manifolds. We provide a simple-tocompute measure of this meta-dependence using moment projections, with a closed-form expression for multivariate Gaussian distributions, and consolidate our findings empirically using both synthetic and real-world data. Our measure of metadependence does not rely on graphical properties of the distribution and can be computed directly from summary statistics such as a covariance matrix, allowing for various applications. We demonstrate one use case of meta-dependence, using a simple redundancy metric to tune significance thresholds and improve causal discovery.  \n1 INTRODUCTION  \nStructural Causal Models (SCMs), popularized by Pearl [1998], are networks of causal dependencies that drive datagenerating processes. Knowing the graphical structure of these networks enables us to identify causal effects [Pearl, 2009], efficiently search for root causes [Ikram et al., 2022], and design more informative experiments [Zhang et al., 2023] . In practice, however, the SCM is rarely available  \na priori and must be inferred from data. This is known as causal discovery [Spirtes et al., 2000] .  \nData generated from SCMs exhibit predictable conditional independence (CI) structures from the causal Markov condition, and these structures can be read off the causal graph. For example, a chain A → B → C exhibits a Markov property whereby the first and last random variables are independent given the middle one. D-separation conditions formalize exactly how graphical structure constrains the joint distribution with respect to conditional independence, and the resulting properties are the machinery underlying most graphical methods for causal identification [Pearl, 2009] . The same reasoning underlies methods well beyond effect identification, from invariant feature selection [RojasCarulla et al., 2018] to the transportability of causal and statistical findings across settings [Pearl and Bareinboim, 2014] .  \nMotivation Conditional independence properties are not separate degrees of freedom within a causal system. The“graphoid axioms” [Pearl and Paz, 1986] and the “Verma constraints” [Verma and Pearl, 1990](the latter arising with latent variables) capture how CIs constrain one another, but these are qualitative, graph-level rules. Such dependence matters for any algorithm that uses multiple CI tests: if an algorithm fails when at least one test is incorrect, independent failures are costlier than co-occurring ones. This is most stark in hypothesis testing, where the standard Bonferroni correction treats test errors as independent, overcorrecting when they are redundant. Yet no metric quantifies the information shared between CI tests at the level of the distribution, let alone one computable without the causal graph. We provide a geometric insight into this dependence and a simple-to-compute metric for it.  \n1.1 PROBLEM A","cbCaivSeLgObR3KH","https://ap.wps.com/l/cbCaivSeLgObR3KH","pdf",775122,1,14,"English","en",105,"# Introduction\n## Motivation\n## Problem and Summary","[{\"question\":\"What does “meta-dependence” mean in conditional independence testing?\",\"answer\":\"Meta-dependence captures how outcomes of multiple conditional independence (CI) tests can correlate, especially across repeated subsampling. It quantifies the dependence between CI properties at the distribution level rather than treating test errors as independent.\"},{\"question\":\"How is the proposed meta-dependence measured without the causal graph or raw data?\",\"answer\":\"The method uses only summary statistics of the distribution, such as a covariance matrix. It applies moment projections from an empirical distribution onto manifolds corresponding to CI properties to compute a measure based on changes in KL divergence.\"},{\"question\":\"Why are the relationships between CI tests sometimes positive or negative?\",\"answer\":\"The paper shows that the Conditional Independence Meta-Dependence (CIMD) can be positive or negative, even within models that share the same causal structure. Differences in the structural equations can drive these variations.\"}]",1784195574,35,{"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},"meta-dependence-in-conditional-independence-testing","",{"@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/meta-dependence-in-conditional-independence-testing/84428/",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 does “meta-dependence” mean in conditional independence testing?","Question",{"text":75,"@type":76},"Meta-dependence captures how outcomes of multiple conditional independence (CI) tests can correlate, especially across repeated subsampling. It quantifies the dependence between CI properties at the distribution level rather than treating test errors as independent.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the proposed meta-dependence measured without the causal graph or raw data?",{"text":80,"@type":76},"The method uses only summary statistics of the distribution, such as a covariance matrix. It applies moment projections from an empirical distribution onto manifolds corresponding to CI properties to compute a measure based on changes in KL divergence.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are the relationships between CI tests sometimes positive or negative?",{"text":84,"@type":76},"The paper shows that the Conditional Independence Meta-Dependence (CIMD) can be positive or negative, even within models that share the same causal structure. Differences in the structural equations can drive these variations.","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"]