Bayesian network

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)iIisarandomvariablerepresentedbyanodeiinitsdirectedacyclicgraph,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,ifyouwanttostore10variableswhosevalues​​areall0or1asaconditionalprobabilitytabletype,anintuitiveideaknowsthatwehavetocalculateatotalofvalues;butifthereisnocorrelationamongthese10variables."Ifthe“dependent”variableismorethanthreeormore,thentheconditionalprobabilitytableoftheBayesiannetworkonlyneedstocalculateatmostonevalue.AnotheradvantageoftheBayesianInternetisthatitiseasierforhumanstoknowwhetherthevariablesareconditionallyindependentordependentandthetypeoflocaldistribution(localdistribution)tofindallrandomvariablesThejointdistribution.

Solutionmethod

TheaboveexampleisaverysimpleBayesiannetworkmodel,butifthemodelisverycomplex,thentheenumerationmethodwillbeusedtosolvetheprobability.Itbecomesverycomplicatedanddifficulttocalculate,sootheralternativemethodsmustbeused.Generallyspeaking,Bayesianprobabilitycanbecalculatedinthefollowingways:

Precisereasoning

·

Enumeratedreasoningmethod(suchastheaboveexample)

·

·

Variableeliminationalgorithm(variableelimination)

·

Randomreasoning(MonteCarlomethod)

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Directsamplingalgorithm

·

·

Rejectsamplingalgorithm

·

·

Likewiseweightingalgorithm

·

·

MarkovchainMonteCarloMarkovchainMonteCarloalgorithm

·

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:

StructureBayesian network

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.

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