Introduction
Bayesiannetwork,alsoknownasbeliefnetwork,isanextensionofBayesianmethodandiscurrentlyoneofthemosteffectivetheoreticalmodelsinthefieldofuncertainknowledgeexpressionandreasoning.SinceitwasproposedbyPearlin1988,ithasbecomearesearchhotspotinrecentyears.ABayesiannetworkisaDirectedAcyclicGraph(DAG),whichiscomposedofnodesrepresentingvariablesanddirectededgesconnectingthesenodes.Nodesrepresentrandomvariables,andthedirectededgesbetweennodesrepresentthemutualrelationshipbetweennodes(fromtheparentnodetoitschildnodes).Conditionalprobabilityisusedtoexpressthestrengthoftherelationship.Ifthereisnoparentnode,thepriorprobabilityisused.Informationexpression.Thenodevariablecanbeanabstractionofanyproblem,suchas:testvalue,observationphenomenon,opinionconsultation,etc.Itissuitableforexpressingandanalyzinguncertainandprobabilisticevents,appliedtodecision-makingthatconditionallyreliesonmultiplecontrolfactors,andcanmakeinferencesfromincomplete,inaccurateoruncertainknowledgeorinformation.
Математическа дефиниция
НекаG=(I,E)означаванасочена циклична графика( DAG), къдетоIпредставлява набора от всички дези в графиката, аEпредставлява набора от сегменти на насочени свързващи линии и НекаX=(X i)i∈Iе случайна променлива, представена от анод< i>iв неговата насочена циклична графика, ако съвместното разпределение на вероятностите на възелаXможе да бъде изразено като:
тогава X се нарича байесианска мрежа по отношение на насочена циклична графаG, която представлява nodei"причината".
Foranyrandomvariable,thejointdistributioncanbeobtainedbymultiplyingtherespectivelocalconditionalprobabilitydistributions:
Accordingtotheaboveformula,wecancombinethejointdistributionofaBayesiannetworkTheprobabilitydistributioniswrittenas:
(За всяка "зависима" променливаXjспрямо Xi)
Thedifferencebetweentheabovetwoexpressionsliesinthepartoftheconditionalprobability.IntheBayesiannetwork,ifthe"dependent"variableisknown,somenodeswillbeconditionallyindependentfromthe"dependent"variable,andonlyrelatedtothe"dependent"variable.Onlythenodeoftheconditionalprobabilityexists.
Ifthenumberofdependenciesofthejointdistributionisveryrare,usingtheBayesianfunctionmethodcansaveconsiderablememorycapacity.Forexample,ifyouwanttostore10variableswhosevaluesareall0or1asaconditionalprobabilitytabletype,anintuitiveideaknowsthatwehavetocalculateatotalofvalues;butifthereisnocorrelationamongthese10variables."Ifthe“dependent”variableismorethanthreeormore,thentheconditionalprobabilitytableoftheBayesiannetworkonlyneedstocalculateatmostonevalue.AnotheradvantageoftheBayesianInternetisthatitiseasierforhumanstoknowwhetherthevariablesareconditionallyindependentordependentandthetypeoflocaldistribution(localdistribution)tofindallrandomvariablesThejointdistribution.
Solutionmethod
TheaboveexampleisaverysimpleBayesiannetworkmodel,butifthemodelisverycomplex,thentheenumerationmethodwillbeusedtosolvetheprobability.Itbecomesverycomplicatedanddifficulttocalculate,sootheralternativemethodsmustbeused.Generallyspeaking,Bayesianprobabilitycanbecalculatedinthefollowingways:
Прецизно разсъждение
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Изброен метод на разсъждение (като горния пример)
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Алгоритъм за елиминиране на променливи (елиминиране на променливи)
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Случайно разсъждение (метод на МонтеКарло)
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Алгоритъм за директно вземане на проби
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Отхвърля алгоритъм за вземане на проби
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По същия начин алгоритъм за претегляне
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MarkovchainMonteCarloMarkovchainMonteCarloalgorithm
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Here,taketheMarkovchainMonteCarloalgorithmasanexample,andthetypeofMarkovchainMonteCarloalgorithmTherearemany,soonlyoneofthestepsofGibbssamplingisexplainedhere:First,fixthevariablewithagivenvalue,andthenrandomlygiveaninitialvaluetotheothervariableswithoutagivenvalue,andthenenterthefollowingiterativesteps:
(1)Изберете произволно една от променливите без дадена стойност
(2)Вземете примерна нова стойност от условното разпределение и след това преизчислете
Aftertheiterativeclumps,deletethepreviousnumbersthatarenotyetstable,andyoucanfindtheapproximateconditionalprobabilitydistribution.TheadvantageoftheMarkovchainMonteCarloalgorithmisthatitisveryefficientwhencomputingalargenetwork,butthedisadvantageisthattheextractedsamplesarenotindependent.
WhenthestructureandparametersontheBayesiannetworkareknown,wecanusetheabovemethodstofindtheprobabilityofaspecificsituation,butifthestructureorparametersontheInternetareunknown,wemustItismoredifficulttoestimatethestructureorparametersofthenetworkbasedontheobserveddata.Generallyspeaking,itismoredifficulttoestimatethestructureofthenetworkthantheparametersonthenode.AccordingtotheunderstandingoftheBayesiannetworkstructureandthecompletenessoftheobservations,wecandivideitintothefollowingfoursituations:
Структура | Наблюдения | Методи |
Известно | Завършено | Метод за оценка на максималната вероятност (MLE) |
Известно | Част | EMалгоритъм Метод за катерене на GreedyHill |
Неизвестен | Завършено | Търсене в цялото моделно пространство |
Неизвестен | Част | Структурен алгоритъм EMалгоритъм Ограничено свиване |
Features
1.Bayesiannetworkitselfisanuncertaincausalassociationmodel.Bayesiannetworkisdifferentfromotherdecisionmodels.Ititselfisaprobabilisticknowledgeexpressionandreasoningmodelthatvisualizesmultipleknowledgediagrams,anditmorecloselycontainsthecausalrelationshipandconditionalcorrelationbetweennetworknodevariables.
2.Bayesiannetworkhasastrongabilitytodealwithuncertainproblems.Bayesiannetworkexpressesthecorrelationbetweenvariousinformationelementswithconditionalprobability,andcanlearnandreasonundertheconditionoflimited,incompleteanduncertaininformation.
3.Bayesiannetworkscaneffectivelyexpressandintegratemulti-sourceinformation.Bayesiannetworkcanincorporatevariousinformationrelatedtofaultdiagnosisandmaintenancedecision-makingintothenetworkstructure,andprocessitinaunifiedmanneraccordingtothenode,whichcaneffectivelyintegrateinformationrelatedtotherelationship.
ForBayesiannetworkreasoningresearch,avarietyofapproximatereasoningalgorithmsareproposed,whicharemainlydividedintotwocategories:simulation-basedmethodsandsearch-basedmethods.Inthefieldoffaultdiagnosis,asfarasourhydropowersimulationisconcerned,theprobabilityoffailureisoftenverysmall,soitisgenerallymoresuitabletousesearchinferencealgorithms.Foranexample,wemustfirstanalyzewhichalgorithmmodeltouse:
a.)Ifthenodereliabilitynetworkofthisexampleisasimpledirectedgraphstructure,anditsnumberofnodesissmall,AdopttheprecisereasoningofBayesiannetwork,whichincludesmulti-treepropagationalgorithm,clumptreepropagationalgorithm,graphreductionalgorithm,selecttheappropriatealgorithmfortheinstanceevent;
b.)IfitistheinstanceThegraphstructureofthedrawnnodeiscomplexandthenumberofnodesislarge.Wecanuseapproximatereasoningalgorithmtostudyit.Forspecificimplementation,itisbesttosimplifythecomplexandhugenetwork,andthenconsideritincombinationwithprecisereasoning.
Indailylife,peopleoftenmakecommonsensereasoning,andthiskindofreasoningisusuallyinaccurate.Forexample,ifyouseeapersonwithdamphaircominginandyouthinkitisrainingoutside,thenyoumaybewrong;ifyouseeamanandawomanwithachildinthepark,youthinktheyareafamily,youmayalsoMadeamistake.Inengineering,wealsoneedtomakescientificandreasonablereasoning.However,theproblemsinengineeringpracticearegenerallymorecomplicated,andtherearemanyuncertainfactors.Thisbringsgreatdifficultiestoaccuratereasoning.Longago,uncertaintyreasoningwasanimportantresearchfieldofartificialintelligence.Althoughmanyresearchersinthefieldofartificialintelligenceintroduceothernon-probabilisticprinciples,theyalsobelievethatitispossibletoconstructanduseprobabilisticmethodsbasedoncommonsensereasoning.Inordertoimprovetheaccuracyofreasoning,peopleintroducedprobabilitytheory.TheBayesianNetwork(BayesianNetwork)firstproposedbyJudeaPearlin1988isessentiallyaprobability-baseduncertaintyreasoningnetwork.Itisagraphicalmodelusedtoexpresstheconnectionprobabilityofasetofvariables,anditprovidesawaytoexpresscausalinformation.Atthattime,itwasmainlyusedtodealwithuncertaininformationinartificialintelligence.Subsequently,itgraduallybecamethemainstreamofinformationtechnologytodealwithuncertainty,andithasbeenimportantlyappliedinmanyintelligentsystemsinthefieldsofcomputerintelligencescience,industrialcontrol,andmedicaldiagnosis.
Bayesiantheoryisanimportanttooltodealwithuncertaininformation.Asamethodofuncertaintyreasoningbasedonprobability,Bayesiannetworkshavebeenimportantapplicationsinintelligentsystemsdealingwithuncertaininformation,andhavebeensuccessfullyusedinmedicaldiagnosis,statisticaldecision-making,expertsystems,learningpredictions,etc.field.ThesesuccessfulapplicationsfullydemonstratethatBayesiannetworktechnologyisapowerfulmethodofuncertaintyreasoning.
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