# Bayesian network

## Mathematicaldefinition

Foranyrandomvariable,thejointdistributioncanbeobtainedbymultiplyingtherespectivelocalconditionalprobabilitydistributions:

Accordingtotheaboveformula,wecancombinethejointdistributionofaBayesiannetworkTheprobabilitydistributioniswrittenas:

(Foreach"dependent"variableXjrelativetoXi)

Thedifferencebetweentheabovetwoexpressionsliesinthepartoftheconditionalprobability.IntheBayesiannetwork,ifthe"dependent"variableisknown,somenodeswillbeconditionallyindependentfromthe"dependent"variable,andonlyrelatedtothe"dependent"variable.Onlythenodeoftheconditionalprobabilityexists.

## 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

WhenthestructureandparametersontheBayesiannetworkareknown,wecanusetheabovemethodstofindtheprobabilityofaspecificsituation,butifthestructureorparametersontheInternetareunknown,wemustItismoredifficulttoestimatethestructureorparametersofthenetworkbasedontheobserveddata.Generallyspeaking,itismoredifficulttoestimatethestructureofthenetworkthantheparametersonthenode.AccordingtotheunderstandingoftheBayesiannetworkstructureandthecompletenessoftheobservations,wecandivideitintothefollowingfoursituations:

 Structure Observations Methods Known Complete MaximumlikelihoodestimationMethod(MLE) Known Part EMalgorithmGreedyHill-climbingmethod Unknown Complete Searchtheentiremodelspace Unknown Part StructuralalgorithmEMalgorithmBoundcontraction

## 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: