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.
Mathematicaldefinition
LetG=(I,E)denoteaDirectedacyclicgraph(DAG),whereIrepresentsthesetofallnodesinthegraph,andErepresentsthesetofdirectedconnectinglinesegments,andLetX=(Xi)i∈Iisarandomvariablerepresentedbyanodeiinitsdirectedacyclicgraph,IfthejointprobabilitydistributionofthenodeXcanbeexpressedas:
thenXiscalledaBayesiannetworkrelativetoadirectedacyclicgraphG,whichrepresentsnodeiThe"cause".
Foranyrandomvariable,thejointdistributioncanbeobtainedbymultiplyingtherespectivelocalconditionalprobabilitydistributions:
Accordingtotheaboveformula,wecancombinethejointdistributionofaBayesiannetworkTheprobabilitydistributioniswrittenas:
(Foreach"dependent"variableXjrelativetoXi)
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:
Precisereasoning
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Enumeratedreasoningmethod(suchastheaboveexample)
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Variableeliminationalgorithm(variableelimination)
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Randomreasoning(MonteCarlomethod)
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Directsamplingalgorithm
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Rejectsamplingalgorithm
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Likewiseweightingalgorithm
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MarkovchainMonteCarloMarkovchainMonteCarloalgorithm
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Here,taketheMarkovchainMonteCarloalgorithmasanexample,andthetypeofMarkovchainMonteCarloalgorithmTherearemany,soonlyoneofthestepsofGibbssamplingisexplainedhere:First,fixthevariablewithagivenvalue,andthenrandomlygiveaninitialvaluetotheothervariableswithoutagivenvalue,andthenenterthefollowingiterativesteps:
(1)Randomlyselectoneofthevariableswithoutagivenvalue
(2)Sampleanewvaluefromtheconditionaldistribution,andthenrecalculate
Aftertheiterativeclumps,deletethepreviousnumbersthatarenotyetstable,andyoucanfindtheapproximateconditionalprobabilitydistribution.TheadvantageoftheMarkovchainMonteCarloalgorithmisthatitisveryefficientwhencomputingalargenetwork,butthedisadvantageisthattheextractedsamplesarenotindependent.
WhenthestructureandparametersontheBayesiannetworkareknown,wecanusetheabovemethodstofindtheprobabilityofaspecificsituation,butifthestructureorparametersontheInternetareunknown,wemustItismoredifficulttoestimatethestructureorparametersofthenetworkbasedontheobserveddata.Generallyspeaking,itismoredifficulttoestimatethestructureofthenetworkthantheparametersonthenode.AccordingtotheunderstandingoftheBayesiannetworkstructureandthecompletenessoftheobservations,wecandivideitintothefollowingfoursituations:
Structure | Observations | Methods |
Known | Complete | MaximumlikelihoodestimationMethod(MLE) |
Known | Part | EMalgorithm GreedyHill-climbingmethod |
Unknown | Complete | Searchtheentiremodelspace |
Unknown | Part | Structuralalgorithm EMalgorithm Boundcontraction |
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.
TheapplicationlevelofBayesiannetworks
Bayesiannetworksarecurrentlyusedincomputationalbiologyandbioinformaticsgeneregulatorynetworks),proteinstructure,geneexpressionanalysis,medicine,documentclassification,informationretrieval,decisionsupportsystems,engineering),gamingandlaw,datafusion,imageprocessing,etc.