[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-291175-105":59,"doc-detail-291175-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","lazyiter-a-fast-algorithm-for-counting-markov-equivalent-dags-and-designing-experiments","LazyIter - A Fast Algorithm for Counting Markov Equivalent DAGs and Designing Experiments","","Causal relationships among random variables can be represented by directed acyclic graphs (DAGs), but observational data identify the true structure only up to a Markov equivalence class (MEC), whose size reflects the difficulty of recovering the causal graph under interventions. The work introduces LazyIter, enabling efficient iteration over candidate MECs from intervention outcomes. It computes MEC sizes and supports both active and passive experiment design. On sparse graphs, the method reduces time complexity by a factor of O(n) versus prior MEC-counting approaches, and achieves better performance in experiments.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/lazyiter-a-fast-algorithm-for-counting-markov-equivalent-dags-and-designing-experiments/291175/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/lazyiter-a-fast-algorithm-for-counting-markov-equivalent-dags-and-designing-experiments/291175.png","ImageObject",300,407,{"name":92,"@type":93},"Aurelia","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the size of a Markov equivalence class (MEC) measure in causal discovery?","Question",{"text":112,"@type":113},"It measures the complexity of recovering the true causal graph by performing interventions, since observational data identify the structure only up to the MEC.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is LazyIter designed to do?",{"text":117,"@type":113},"LazyIter efficiently iterates over possible MECs that may result after an intervention on a single node by reusing similarities between candidate graphs.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the proposed method improve computation efficiency?",{"text":121,"@type":113},"It reduces the time complexity for computing MEC sizes by a factor of O(n) for sparse graphs compared with previous MEC-counting algorithms.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},291175,1789643596,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":44,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143,"preview_page_count":4},1099514068365,"https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1794309523707217728","LazyIter: A Fast Algorithm for Counting Markov Equivalent DAGs and  \nDesigning Experiments  \nAli AhmadiTeshnizi 1 Saber Salehkaleybar 1 Negar Kiyavash 2  \nAbstract  \nThe causal relationships among a set of random variables are commonly represented by a Directed Acyclic Graph (DAG), where there is a directed edge from variable X to variable Y if X is a direct cause of Y. From the purely observational data, the true causal graph can be identified up to a Markov Equivalence Class (MEC), which is a set of DAGs with the same conditional independencies between the variables. The size of an MEC is a measure of complexity for recovering the true causal graph by performing interventions.  \nWe propose a method for efficient iteration over possible MECs given intervention results. We utilize the proposed method for computing MEC sizes and experiment design in active and passive learning settings. Compared to previous work for computing the size of MEC, our proposed algorithm reduces the time complexity by a factor of O (n) for sparse graphs where n is the number of variables in the system. Additionally, integrating our approach with dynamic programming, we design an optimal algorithm for passive experiment design. Experimental results show that our proposed algorithms for both computing the size of MEC and experiment design outperform the state of the art.  \n1. Introduction  \nDirected Acyclic Graphs (DAGs) are the most commonly used structures to represent causal relations between random variables, where a directed edge X → Y means that  \n1Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran 2 School of Management of Technology, ´Ecole Polytechnique Fdrale de Lausanne, Switzerland. Correspondence to: Ali AhmadiTeshnizi \u003C[ali.ahmadi215@student.sharif.edu](ali.ahmadi215@student.sharif.edu)>, Saber Salehkaleybar (Corresponding author) \u003C[saleh@sharif.edu](saleh@sharif.edu)>, Negar Kiyavash \u003Cnegar.kiyavash@epfl.ch> .  \nProceedings of the 37th International Conference on Machine Learning, Online, PMLR 119, 2020 . Copyright 2020 by the author(s) .  \nvariable X is a direct cause of variable Y. Under the faithfulness assumption, conditional independencies between different variables can be inferred from observational data and consequently, the ground truth graph is identified up to Markov Equivalence Class (MEC) (Pearl, 2009 ; Spirteset al., 2000) . Unique identification of the ground truth DAG among the graphs in an MEC generally requires interventions on variables (Eberhardt & Scheines, 2007) . In some scenarios, interventions could be costly (for instance, in biological experiments), and therefore, selecting the optimal intervention target to learn the causal structure is of great interest (Eberhardt et al., 2005 ; He & Geng, 2008 ; Eberhardt, 2008 ; Hauser & B¨uhlmann, 2014 ; Shanmugamet al., 2015 ; Kocaoglu et al., 2017 ; Ghassami et al., 2018 ; Lindgren et al., 2018 ; Agrawal et al., 2019) . Several metricshave been suggested in the literature for target selection (He & Geng, 2008 ; Hauser & B¨uhlmann, 2014 ; Ghassami et al., 2018 ; Agrawal et al., 2019) . A good metric for measuring the effectiveness of an intervention is the number of remaining DAGs in an MEC after the intervention (He & Geng, 2008) . To use this metric for target selection, we must be able to efficiently count the number of DAGs in an MEC.  \nSome previous work used the clique tree representation of chordal graphs to divide the causal graph into smaller subgraphs, and perform counting on each subgraph separately (Ghassami et al., 2019 ; Talvitie & Koivisto, 2019) . The main issue with this approach is the dependence on the maximum clique size which can result in O (n!) operations in some cases where n is the number of variables. Ghassamiet al. (2019) and Talvitie & Koivisto (2019) used dynamic programming to count DAGs in an MEC. The best time complexity of these approaches is in the order of O(2nn4 ) . However, both approaches do not take advanta","cbCaicIgwylQaaBb","https://ap.wps.com/l/cbCaicIgwylQaaBb","pdf",343992,"English","# Abstract\n# 1. Introduction","[{\"question\":\"What does the size of a Markov equivalence class (MEC) measure in causal discovery?\",\"answer\":\"It measures the complexity of recovering the true causal graph by performing interventions, since observational data identify the structure only up to the MEC.\"},{\"question\":\"What is LazyIter designed to do?\",\"answer\":\"LazyIter efficiently iterates over possible MECs that may result after an intervention on a single node by reusing similarities between candidate graphs.\"},{\"question\":\"How does the proposed method improve computation efficiency?\",\"answer\":\"It reduces the time complexity for computing MEC sizes by a factor of O(n) for sparse graphs compared with previous MEC-counting algorithms.\"}]","LazyIter - A Fast Algorithm for Counting Markov Equivalent DAGs and Designing Experiments | PDF",23]