{"id":"https://openalex.org/W2174375947","doi":"https://doi.org/10.1109/tit.2013.2267934","title":"Universal Estimation of Directed Information","display_name":"Universal Estimation of Directed Information","publication_year":2013,"publication_date":"2013-07-09","ids":{"openalex":"https://openalex.org/W2174375947","doi":"https://doi.org/10.1109/tit.2013.2267934","mag":"2174375947"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2013.2267934","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2013.2267934","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1201.2334","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034192173","display_name":"Jiantao Jiao","orcid":"https://orcid.org/0000-0003-3766-8031"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jiantao Jiao","raw_affiliation_strings":["Department of Electrical Engineering, Stanford University, Stanford, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Stanford University, Stanford, CA, USA","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068879871","display_name":"Haim H. Permuter","orcid":"https://orcid.org/0000-0003-3170-3190"},"institutions":[{"id":"https://openalex.org/I124227911","display_name":"Ben-Gurion University of the Negev","ror":"https://ror.org/05tkyf982","country_code":"IL","type":"education","lineage":["https://openalex.org/I124227911"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Haim H. Permuter","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beersheba, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beersheba, Israel","institution_ids":["https://openalex.org/I124227911"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042644835","display_name":"Lei Zhao","orcid":"https://orcid.org/0000-0002-6551-2707"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lei Zhao","raw_affiliation_strings":["Stanford University, Stanford, CA, US"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Stanford University, Stanford, CA, US","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101647166","display_name":"Young-Han Kim","orcid":"https://orcid.org/0000-0001-7255-8757"},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Young-Han Kim","raw_affiliation_strings":["Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA, USA","institution_ids":["https://openalex.org/I36258959"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043344688","display_name":"Tsachy Weissman","orcid":"https://orcid.org/0009-0008-1099-691X"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tsachy Weissman","raw_affiliation_strings":["Department of Electrical Engineering, Stanford University, Stanford, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Stanford University, Stanford, CA, USA","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":12.8474,"has_fulltext":true,"cited_by_count":120,"citation_normalized_percentile":{"value":0.98615346,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"59","issue":"10","first_page":"6220","last_page":"6242"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11447","display_name":"Blind Source Separation Techniques","score":0.9970999956130981,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.8377587199211121},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.637564480304718},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.612606406211853},{"id":"https://openalex.org/keywords/minimax","display_name":"Minimax","score":0.5814599990844727},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.5473939180374146},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5319894552230835},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.4777349829673767},{"id":"https://openalex.org/keywords/mutual-information","display_name":"Mutual information","score":0.4766072630882263},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4224506616592407},{"id":"https://openalex.org/keywords/probability-distribution","display_name":"Probability distribution","score":0.41185706853866577},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4080972969532013},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.38980284333229065},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3710324764251709},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.2069319188594818}],"concepts":[{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.8377587199211121},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.637564480304718},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.612606406211853},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.5814599990844727},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.5473939180374146},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5319894552230835},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.4777349829673767},{"id":"https://openalex.org/C152139883","wikidata":"https://www.wikidata.org/wiki/Q252973","display_name":"Mutual information","level":2,"score":0.4766072630882263},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4224506616592407},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.41185706853866577},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4080972969532013},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.38980284333229065},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3710324764251709},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2069319188594818},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C127162648","wikidata":"https://www.wikidata.org/wiki/Q16858953","display_name":"Channel (broadcasting)","level":2,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tit.2013.2267934","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2013.2267934","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1201.2334","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1201.2334","pdf_url":"https://arxiv.org/pdf/1201.2334","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.708.2298","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.708.2298","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://circuit.ucsd.edu/%7Eyhk/pdfs/it2013oct.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.762.4724","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.762.4724","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1201.2334.pdf","raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1201.2334","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1201.2334","pdf_url":"https://arxiv.org/pdf/1201.2334","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1523888516","display_name":null,"funder_award_id":"FA9550-","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G1739869781","display_name":null,"funder_award_id":"CCF-0939370","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G2568697414","display_name":null,"funder_award_id":"CCF-0939370","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G5809100787","display_name":null,"funder_award_id":"FA9550","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G6244900323","display_name":null,"funder_award_id":"0939370","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2174375947.pdf","grobid_xml":"https://content.openalex.org/works/W2174375947.grobid-xml"},"referenced_works_count":49,"referenced_works":["https://openalex.org/W171065245","https://openalex.org/W1489262227","https://openalex.org/W1511694993","https://openalex.org/W1549664537","https://openalex.org/W1591308451","https://openalex.org/W1742030380","https://openalex.org/W1968250327","https://openalex.org/W1970091448","https://openalex.org/W1982389390","https://openalex.org/W1997136004","https://openalex.org/W1999120268","https://openalex.org/W2012639032","https://openalex.org/W2012792046","https://openalex.org/W2013610892","https://openalex.org/W2018640312","https://openalex.org/W2024886389","https://openalex.org/W2035364928","https://openalex.org/W2082484220","https://openalex.org/W2083048215","https://openalex.org/W2091891040","https://openalex.org/W2100352482","https://openalex.org/W2100967164","https://openalex.org/W2103756638","https://openalex.org/W2110764329","https://openalex.org/W2116208397","https://openalex.org/W2126785155","https://openalex.org/W2129058877","https://openalex.org/W2129885443","https://openalex.org/W2129902114","https://openalex.org/W2133648300","https://openalex.org/W2134392502","https://openalex.org/W2136010267","https://openalex.org/W2140911775","https://openalex.org/W2156507576","https://openalex.org/W2156939381","https://openalex.org/W2157721994","https://openalex.org/W2161488606","https://openalex.org/W2163294786","https://openalex.org/W2165397914","https://openalex.org/W2168614460","https://openalex.org/W2170452707","https://openalex.org/W2178225550","https://openalex.org/W2183235533","https://openalex.org/W2294584261","https://openalex.org/W2478708596","https://openalex.org/W2797937785","https://openalex.org/W4245724425","https://openalex.org/W4294566064","https://openalex.org/W6637815268"],"related_works":["https://openalex.org/W50969306","https://openalex.org/W2390585021","https://openalex.org/W412641959","https://openalex.org/W2341040961","https://openalex.org/W4287207389","https://openalex.org/W2779828239","https://openalex.org/W3154976382","https://openalex.org/W4292101436","https://openalex.org/W2044004505","https://openalex.org/W2558128491"],"abstract_inverted_index":{"Four":[0],"estimators":[1,77,101,126,168],"of":[2,10,47,65,99,116,135,192,201],"the":[3,53,66,97,119,158,166,172,177,190,193,199],"directed":[4,202],"information":[5,49,133,146,203],"rate":[6,143],"between":[7],"a":[8,27],"pair":[9],"jointly":[11],"stationary":[12,136],"ergodic":[13,137],"finite-alphabet":[14,138],"processes":[15],"are":[16,169,187],"proposed,":[17],"based":[18],"on":[19,182],"universal":[20,71,79,178],"probability":[21,72,80,179],"assignments.":[22],"The":[23,74],"first":[24],"one":[25],"is":[26],"Shannon\u2013McMillan\u2013Breiman-type":[28],"estimator,":[29],"similar":[30],"to":[31,82,130,154],"those":[32],"used":[33],"by":[34,162],"Verd\u00fa":[35],"in":[36,43,102,118,157,196,205],"2005":[37],"and":[38,56,93,105,112,144,151,184,198,207,211],"Cai":[39],"<etal":[40],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[41,59,108],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"/>":[42],"2006":[44],"for":[45,68],"estimation":[46,204],"other":[48,75,132],"measures.":[50],"We":[51,95],"show":[52],"almost":[54,103],"sure":[55,104],"<formula":[57,106],"formulatype=\"inline\"":[58,107],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><tex":[60,109],"Notation=\"TeX\">$L_{1}$</tex>":[61],"</formula>":[62],"convergence":[63,117],"properties":[64],"estimator":[67],"any":[69],"underlying":[70],"assignment.":[73,180],"three":[76],"map":[78],"assignments":[81],"different":[83],"functionals,":[84],"each":[85],"exhibiting":[86],"relative":[87],"merits":[88],"such":[89,140],"as":[90,141,176],"smoothness,":[91],"nonnegativity,":[92],"boundedness.":[94],"establish":[96],"consistency":[98],"these":[100,163],"Notation=\"TeX\">$L_{1}$</tex></formula>":[110],"senses,":[111],"derive":[113],"near-optimal":[114,149],"rates":[115],"minimax":[120],"sense":[121],"under":[122],"mild":[123],"conditions.":[124],"These":[125],"carry":[127],"over":[128],"directly":[129],"estimating":[131],"measures":[134],"processes,":[139],"entropy":[142],"mutual":[145],"rate,":[147],"with":[148],"performance":[150],"provide":[152],"alternatives":[153],"classical":[155],"approaches":[156],"existing":[159],"literature.":[160],"Guided":[161],"theoretical":[164],"results,":[165],"proposed":[167,194],"implemented":[170],"using":[171],"context-tree":[173],"weighting":[174],"algorithm":[175],"Experiments":[181],"synthetic":[183],"real":[185],"data":[186],"presented,":[188],"demonstrating":[189],"potential":[191],"schemes":[195],"practice":[197],"utility":[200],"detecting":[206],"measuring":[208],"causal":[209],"influence":[210],"delay.":[212]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":13},{"year":2019,"cited_by_count":15},{"year":2018,"cited_by_count":17},{"year":2017,"cited_by_count":11},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}