{"id":"https://openalex.org/W2158257766","doi":"https://doi.org/10.1109/tit.1970.1054456","title":"Likelihood ratios for Gaussian processes","display_name":"Likelihood ratios for Gaussian processes","publication_year":1970,"publication_date":"1970-05-01","ids":{"openalex":"https://openalex.org/W2158257766","doi":"https://doi.org/10.1109/tit.1970.1054456","mag":"2158257766"},"language":"en","primary_location":{"id":"doi:10.1109/tit.1970.1054456","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1970.1054456","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":["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/A5014088470","display_name":"T. Kailath","orcid":null},"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"]},{"id":"https://openalex.org/I59270414","display_name":"Indian Institute of Science Bangalore","ror":"https://ror.org/04dese585","country_code":"IN","type":"education","lineage":["https://openalex.org/I59270414"]}],"countries":["IN","US"],"is_corresponding":true,"raw_author_name":"T. Kailath","raw_affiliation_strings":["Stanford University, Stanford, CA, USA","Department of Electrical Communication Engineering, Indian Institute of Technology, Bangalore, India","University of Stanford, Stanford, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Stanford University, Stanford, CA, USA","institution_ids":["https://openalex.org/I97018004"]},{"raw_affiliation_string":"Department of Electrical Communication Engineering, Indian Institute of Technology, Bangalore, India","institution_ids":["https://openalex.org/I59270414"]},{"raw_affiliation_string":"University of Stanford, Stanford, CA, USA","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5014088470"],"corresponding_institution_ids":["https://openalex.org/I59270414","https://openalex.org/I97018004"],"apc_list":null,"apc_paid":null,"fwci":23.4672,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.99732262,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"16","issue":"3","first_page":"276","last_page":"288"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9886000156402588,"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/T13126","display_name":"Scientific Research and Discoveries","score":0.9868000149726868,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.644832968711853},{"id":"https://openalex.org/keywords/resolvent","display_name":"Resolvent","score":0.5991326570510864},{"id":"https://openalex.org/keywords/white-noise","display_name":"White noise","score":0.595024049282074},{"id":"https://openalex.org/keywords/additive-white-gaussian-noise","display_name":"Additive white Gaussian noise","score":0.5857785940170288},{"id":"https://openalex.org/keywords/gaussian-noise","display_name":"Gaussian noise","score":0.5835886001586914},{"id":"https://openalex.org/keywords/invertible-matrix","display_name":"Invertible matrix","score":0.5447718501091003},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.5055553913116455},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4967723488807678},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.496420681476593},{"id":"https://openalex.org/keywords/gaussian-process","display_name":"Gaussian process","score":0.48953256011009216},{"id":"https://openalex.org/keywords/gaussian-random-field","display_name":"Gaussian random field","score":0.48444536328315735},{"id":"https://openalex.org/keywords/noise","display_name":"Noise (video)","score":0.4180715084075928},{"id":"https://openalex.org/keywords/discrete-time-signal","display_name":"Discrete-time signal","score":0.4159241318702698},{"id":"https://openalex.org/keywords/stochastic-resonance","display_name":"Stochastic resonance","score":0.4144613742828369},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39396345615386963},{"id":"https://openalex.org/keywords/signal-transfer-function","display_name":"Signal transfer function","score":0.3800293207168579},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.31278228759765625},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2772994339466095},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1747458577156067},{"id":"https://openalex.org/keywords/analog-signal","display_name":"Analog signal","score":0.1121896505355835},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10065782070159912}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.644832968711853},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.5991326570510864},{"id":"https://openalex.org/C112633086","wikidata":"https://www.wikidata.org/wiki/Q381287","display_name":"White noise","level":2,"score":0.595024049282074},{"id":"https://openalex.org/C169334058","wikidata":"https://www.wikidata.org/wiki/Q353292","display_name":"Additive white Gaussian noise","level":3,"score":0.5857785940170288},{"id":"https://openalex.org/C4199805","wikidata":"https://www.wikidata.org/wiki/Q2725903","display_name":"Gaussian noise","level":2,"score":0.5835886001586914},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.5447718501091003},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5055553913116455},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4967723488807678},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.496420681476593},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.48953256011009216},{"id":"https://openalex.org/C51267290","wikidata":"https://www.wikidata.org/wiki/Q5527848","display_name":"Gaussian random field","level":4,"score":0.48444536328315735},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.4180715084075928},{"id":"https://openalex.org/C146749787","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete-time signal","level":5,"score":0.4159241318702698},{"id":"https://openalex.org/C207658827","wikidata":"https://www.wikidata.org/wiki/Q1999781","display_name":"Stochastic resonance","level":4,"score":0.4144613742828369},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39396345615386963},{"id":"https://openalex.org/C131021393","wikidata":"https://www.wikidata.org/wiki/Q7512759","display_name":"Signal transfer function","level":4,"score":0.3800293207168579},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.31278228759765625},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2772994339466095},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1747458577156067},{"id":"https://openalex.org/C13412647","wikidata":"https://www.wikidata.org/wiki/Q174948","display_name":"Analog signal","level":3,"score":0.1121896505355835},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10065782070159912},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C9390403","wikidata":"https://www.wikidata.org/wiki/Q3966","display_name":"Computer hardware","level":1,"score":0.0},{"id":"https://openalex.org/C84462506","wikidata":"https://www.wikidata.org/wiki/Q173142","display_name":"Digital signal processing","level":2,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.1970.1054456","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1970.1054456","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"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.550000011920929},{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.4399999976158142}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W136746510","https://openalex.org/W226131373","https://openalex.org/W602200405","https://openalex.org/W1005860193","https://openalex.org/W1480674130","https://openalex.org/W1500551892","https://openalex.org/W1595924086","https://openalex.org/W1973376127","https://openalex.org/W1978990971","https://openalex.org/W2010659807","https://openalex.org/W2012761257","https://openalex.org/W2014831690","https://openalex.org/W2031755322","https://openalex.org/W2035925506","https://openalex.org/W2062973984","https://openalex.org/W2099292835","https://openalex.org/W2103435373","https://openalex.org/W2125398376","https://openalex.org/W2134390549","https://openalex.org/W2162176610","https://openalex.org/W2169653324","https://openalex.org/W2241375259","https://openalex.org/W2321957512","https://openalex.org/W2529873332","https://openalex.org/W2799137445","https://openalex.org/W3147179697","https://openalex.org/W3175310863","https://openalex.org/W4213062135","https://openalex.org/W4213230606","https://openalex.org/W4250162080","https://openalex.org/W4300514222","https://openalex.org/W6618188213","https://openalex.org/W7064804000"],"related_works":["https://openalex.org/W2018889859","https://openalex.org/W1974988056","https://openalex.org/W3209348608","https://openalex.org/W2385533587","https://openalex.org/W2954053562","https://openalex.org/W2012879029","https://openalex.org/W2142861680","https://openalex.org/W2010201620","https://openalex.org/W2001116605","https://openalex.org/W2024512500"],"abstract_inverted_index":{"We":[0,37],"give":[1],"a":[2,72,121,136,141,157],"comprehensive":[3],"discussion":[4,158],"of":[5,8,20,44,53,64,71,103,123,148,159],"the":[6,9,54,61,65,77,87,100,104,108,160],"structure":[7],"likelihood":[10],"ratio":[11],"(LR)":[12],"for":[13,79,92,144],"discrimination":[14],"between":[15],"two":[16],"Gaussian":[17,97,170],"processes,":[18,146],"one":[19],"which":[21,149],"is":[22],"white.":[23],"Several":[24],"more":[25,56],"general":[26],"problems":[27,43,52,172],"can":[28,47,83],"be":[29,49,70,84,115,118],"reduced,":[30],"usually":[31],"by":[32],"differentiation,":[33],"to":[34,117,120],"this":[35,45,80],"form.":[36,75],"shall":[38],"show":[39],"that":[40,91,167],"nonsingular":[41],"detection":[42,171],"form":[46,89],"always":[48],"interpreted":[50],"as":[51,90],"apparently":[55],"special":[57,73],"\"signal-in-noise\"":[58],"type,":[59],"where":[60],"cross-covariance":[62],"function":[63],"signal":[66,105],"and":[67,139,164],"noise":[68],"must":[69],"\"one-sided\"":[74],"Moreover,":[76],"LR":[78],"equivalent":[81,119],"problem":[82],"written":[85],"in":[86,95,169],"same":[88],"known":[93,109],"signals":[94],"white":[96],"noise,":[98],"with":[99],"causal":[101],"estimate":[102],"process":[106],"replacing":[107],"signal.":[110],"This":[111,153],"single":[112],"formula":[113],"will":[114],"shown":[116],"variety":[122],"other":[124,151],"formulas,":[125],"including":[126],"all":[127],"those":[128],"previously":[129],"known.":[130],"The":[131],"proofs":[132],"are":[133],"based":[134],"on":[135,140],"resolvent":[137],"identity":[138],"representation":[142],"theorem":[143],"second-order":[145],"both":[147],"have":[150],"applications.":[152],"paper":[154],"also":[155],"contains":[156],"various":[161],"stochastic":[162],"integrals":[163],"infinite":[165],"determinants":[166],"arise":[168]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2016,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}