{"id":"https://openalex.org/W1732328119","doi":"https://doi.org/10.1109/isit.2015.7282680","title":"Adaptive estimation of Shannon entropy","display_name":"Adaptive estimation of Shannon entropy","publication_year":2015,"publication_date":"2015-06-01","ids":{"openalex":"https://openalex.org/W1732328119","doi":"https://doi.org/10.1109/isit.2015.7282680","mag":"1732328119"},"language":"en","primary_location":{"id":"doi:10.1109/isit.2015.7282680","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isit.2015.7282680","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2015 IEEE International Symposium on Information Theory (ISIT)","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077972639","display_name":"Yanjun Han","orcid":"https://orcid.org/0000-0002-8335-2364"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yanjun Han","raw_affiliation_strings":["Tsinghua University","#N##TAB##TAB##TAB##TAB# Tsinghua University, China#N##TAB##TAB##TAB#"],"affiliations":[{"raw_affiliation_string":"Tsinghua University","institution_ids":["https://openalex.org/I99065089"]},{"raw_affiliation_string":"#N##TAB##TAB##TAB##TAB# Tsinghua University, China#N##TAB##TAB##TAB#","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","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":["Stanford University","Stanford University - USA >  >  >  >"],"affiliations":[{"raw_affiliation_string":"Stanford University","institution_ids":["https://openalex.org/I97018004"]},{"raw_affiliation_string":"Stanford University - USA >  >  >  >","institution_ids":["https://openalex.org/I97018004"]}]},{"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":["Stanford University","Stanford University - USA >  >  >  >"],"affiliations":[{"raw_affiliation_string":"Stanford University","institution_ids":["https://openalex.org/I97018004"]},{"raw_affiliation_string":"Stanford University - USA >  >  >  >","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5077972639"],"corresponding_institution_ids":["https://openalex.org/I99065089"],"apc_list":null,"apc_paid":null,"fwci":3.366,"has_fulltext":false,"cited_by_count":25,"citation_normalized_percentile":{"value":0.92300989,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"1372","last_page":"1376"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9898999929428101,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9898999929428101,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9740999937057495,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.9624000191688538,"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":0.8056284785270691},{"id":"https://openalex.org/keywords/minimax","display_name":"Minimax","score":0.7497532367706299},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7275763750076294},{"id":"https://openalex.org/keywords/minimax-estimator","display_name":"Minimax estimator","score":0.5693348050117493},{"id":"https://openalex.org/keywords/pointwise","display_name":"Pointwise","score":0.5481359362602234},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5473464131355286},{"id":"https://openalex.org/keywords/sample-size-determination","display_name":"Sample size determination","score":0.5184276103973389},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.45191580057144165},{"id":"https://openalex.org/keywords/adaptive-estimator","display_name":"Adaptive estimator","score":0.4414181113243103},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4408376216888428},{"id":"https://openalex.org/keywords/information-theory","display_name":"Information theory","score":0.41902539134025574},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.389120489358902},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.34268689155578613},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3400728702545166},{"id":"https://openalex.org/keywords/minimum-variance-unbiased-estimator","display_name":"Minimum-variance unbiased estimator","score":0.17143133282661438},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08742204308509827}],"concepts":[{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.8056284785270691},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.7497532367706299},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7275763750076294},{"id":"https://openalex.org/C133939421","wikidata":"https://www.wikidata.org/wiki/Q6865379","display_name":"Minimax estimator","level":4,"score":0.5693348050117493},{"id":"https://openalex.org/C2777984123","wikidata":"https://www.wikidata.org/wiki/Q9248237","display_name":"Pointwise","level":2,"score":0.5481359362602234},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5473464131355286},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.5184276103973389},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.45191580057144165},{"id":"https://openalex.org/C2780617739","wikidata":"https://www.wikidata.org/wiki/Q4680738","display_name":"Adaptive estimator","level":3,"score":0.4414181113243103},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4408376216888428},{"id":"https://openalex.org/C52622258","wikidata":"https://www.wikidata.org/wiki/Q131222","display_name":"Information theory","level":2,"score":0.41902539134025574},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.389120489358902},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.34268689155578613},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3400728702545166},{"id":"https://openalex.org/C165646398","wikidata":"https://www.wikidata.org/wiki/Q3755281","display_name":"Minimum-variance unbiased estimator","level":3,"score":0.17143133282661438},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08742204308509827},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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":1,"locations":[{"id":"doi:10.1109/isit.2015.7282680","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isit.2015.7282680","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2015 IEEE International Symposium on Information Theory (ISIT)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W1482537073","https://openalex.org/W1511694993","https://openalex.org/W1533375765","https://openalex.org/W1568559199","https://openalex.org/W1595409123","https://openalex.org/W1732328119","https://openalex.org/W1856858227","https://openalex.org/W1875247892","https://openalex.org/W1989151402","https://openalex.org/W1995875735","https://openalex.org/W2045740840","https://openalex.org/W2062899906","https://openalex.org/W2083620802","https://openalex.org/W2114771311","https://openalex.org/W2117410972","https://openalex.org/W2117509237","https://openalex.org/W2119757675","https://openalex.org/W2127090196","https://openalex.org/W2148719904","https://openalex.org/W2203041117","https://openalex.org/W2319338832","https://openalex.org/W2963389257","https://openalex.org/W2963608890","https://openalex.org/W4205354563","https://openalex.org/W4245378867","https://openalex.org/W4247267886","https://openalex.org/W4394635572","https://openalex.org/W6637653789","https://openalex.org/W6639353696","https://openalex.org/W6681713883","https://openalex.org/W6687612455"],"related_works":["https://openalex.org/W3174947331","https://openalex.org/W2124749597","https://openalex.org/W4388999687","https://openalex.org/W2964153509","https://openalex.org/W1817823578","https://openalex.org/W4292553467","https://openalex.org/W2043395486","https://openalex.org/W1975384468","https://openalex.org/W2952315891","https://openalex.org/W2782094744"],"abstract_inverted_index":{"We":[0,145,206],"consider":[1],"estimating":[2,40],"the":[3,32,54,60,67,74,83,89,97,103,110,120,135,148,152,164,167,170,181,191,212,224,228,238],"Shannon":[4],"entropy":[5],"of":[6,92,99,128,130,151,169,180,193,211,227,237],"a":[7,125,208],"discrete":[8],"distribution":[9],"P":[10,142],"from":[11],"n":[12,47,78,175,184,186,232,241,243],"i.i.d.":[13],"samples.":[14,81,244],"Recently,":[15],"Jiao,":[16,201],"Venkat,":[17,202],"Han,":[18,203],"and":[19,22,24,57,107,158,204],"Weissman":[20],"(JVHW),":[21],"Wu":[23],"Yang":[25],"constructed":[26],"approximation":[27],"theoretic":[28],"estimators":[29,43],"that":[30,109,166,179,219,236],"achieve":[31],"minimax":[33,90,121,171],"L":[34],"<sub":[35],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[36],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sub>":[37],"rates":[38,122],"in":[39,163],"entropy.":[41],"Their":[42],"are":[44],"consistent":[45],"given":[46],"\u226b":[48,79],"S/lnS":[49],"samples,":[50,187],"where":[51],"S":[52,80,138],"is":[53,59,73,113,177,234],"support":[55,136],"size,":[56],"it":[58,118],"best":[61],"possible":[62],"sample":[63,195,213],"complexity.":[64],"In":[65,82],"contrast,":[66],"Maximum":[68],"Likelihood":[69],"Estimator":[70],"(MLE),":[71],"which":[72,140,217],"empirical":[75],"entropy,":[76],"requires":[77],"present":[84],"paper":[85],"we":[86,101],"significantly":[87],"refine":[88],"results":[91],"existing":[93],"work.":[94],"To":[95],"alleviate":[96],"pessimism":[98],"minimaxity,":[100],"adopt":[102],"adaptive":[104,115],"estimation":[105],"framework,":[106],"show":[108],"JVHW":[111,229],"estimator":[112,173,230],"an":[114],"estimator,":[116],"i.e.,":[117],"achieves":[119],"simultaneously":[123],"over":[124,154],"nested":[126,156],"sequence":[127],"subsets":[129],"distributions":[131],"P,":[132],"without":[133],"knowing":[134],"size":[137,196,214],"or":[139],"subset":[141,162],"lies":[143],"in.":[144],"also":[146],"characterize":[147],"maximum":[149],"risk":[150],"MLE":[153,182,239],"this":[155],"sequence,":[157,165],"show,":[159],"for":[160,220],"every":[161],"performance":[168],"rate-optimal":[172],"with":[174,183,231,240],"samples":[176,233],"essentially":[178],"ln":[185,242],"thereby":[188],"further":[189],"substantiating":[190],"generality":[192],"\u201ceffective":[194],"enlargement\u201d":[197],"phenomenon":[198],"discovered":[199],"by":[200],"Weissman.":[205],"provide":[207],"\u201cpointwise\u201d":[209],"explanation":[210],"enlargement":[215],"phenomenon,":[216],"states":[218],"sufficiently":[221],"small":[222],"probabilities,":[223],"bias":[225],"function":[226],"nearly":[235]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3}],"updated_date":"2026-01-13T01:12:25.745995","created_date":"2025-10-10T00:00:00"}