人工智能贝叶斯网络.ppt

1ArtificialIntelligence:BayesianNetworks2GraphicalModels•Ifnoassumptionofindependenceismade,thenanexponentialnumberofparametersmustbeestimatedforsoundprobabilisticinference.•Norealisticamountoftrainingdataissufficienttoestimatesomanyparameters.•Ifablanketassumptionofconditionalindependenceismade,efficienttrainingandinferenceispossible,butsuchastrongassumptionisrarelywarranted.•Graphicalmodelsusedirectedorundirectedgraphsoverasetofrandomvariablestoexplicitlyspecifyvariabledependenciesandallowforlessrestrictiveindependenceassumptionswhilelimitingthenumberofparametersthatmustbeestimated.–BayesianNetworks:Directedacyclicgraphsthatindicatecausalstructure.–MarkovNetworks:Undirectedgraphsthatcapturegeneraldependencies.3BayesianNetworks•DirectedAcyclicGraph(DAG)–Nodesarerandomvariables–EdgesindicatecausalinfluencesBurglaryEarthquakeAlarmJohnCallsMaryCalls4ConditionalProbabilityTables•Eachnodehasaconditionalprobabilitytable(CPT)thatgivestheprobabilityofeachofitsvaluesgiveneverypossiblecombinationofvaluesforitsparents(conditioningcase).–Roots(sources)oftheDAGthathavenoparentsaregivenpriorprobabilities.BurglaryEarthquakeAlarmJohnCallsMaryCallsP(B).001P(E).002BEP(A)TT.95TF.94FT.29FF.001AP(M)T.70F.01AP(J)T.90F.055CPTComments•Probabilityoffalsenotgivensincerowsmustaddto1.•Examplerequires10parametersratherthan25–1=31forspecifyingthefulljointdistribution.•NumberofparametersintheCPTforanodeisexponentialinthenumberofparents(fan-in).6JointDistributionsforBayesNets•ABayesianNetworkimplicitlydefinesajointdistribution.))(Parents|(),...,(121iniinXxPxxxP•Example)(EBAMJP)()()|()|()|(EPBPEBAPAMPAJP00062.0998.0999.0001.07.09.0•Thereforeaninefficientapproachtoinferenceis:–1)Computethejointdistributionusingthisequation.–2)Computeanydesiredconditionalprobabilityusingthejointdistribution.7NaïveBayesasaBayesNet•NaïveBayesisasimpleBayesNetYX1X2…Xn•PriorsP(Y)andconditionalsP(Xi|Y)forNaïveBayesprovideCPTsforthenetwork.8IndependenciesinBayesNets•IfremovingasubsetofnodesSfromthenetworkrendersnodesXiandXjdisconnected,thenXiandXjareindependentgivenS,i.e.P(Xi|Xj,S)=P(Xi|S)•However,thisistoostrictacriteriaforconditionalindependencesincetwonodeswillstillbeconsideredindependentiftheirsimplyexistssomevariablethatdependsonboth.–Forexample,BurglaryandEarthquakeshouldbeconsideredindependentsincetheybothcauseAlarm.9IndependenciesinBayesNets•IfremovingasubsetofnodesSfromthenetworkrendersnodesXiandXjdisconnected,thenXiandXjareindependentgivenS,i.e.P(Xi|Xj,S)=P(Xi|S)•However,thisistoostrictacriteriaforconditionalindependencesincetwonodeswillstillbeconsideredindependentiftheirsimplyexistssomevariablethatdependsonboth.–Forexample,BurglaryandEarthquakeshouldbeconsideredindependentsincetheybothcauseAlarm.P(Xi|Xj,S)=P(Xi|S),isequivalenttoP(Xi,Xj|S)=P(Xi|S)P(Xj|S)Howtoprove?10IndependenciesinBayesNets•IfremovingasubsetofnodesSfromthenetworkrendersnodesXiandXjdisconnected,thenXiandXjareindependentgivenS,i.e.P(Xi|Xj,S)=P(Xi|S)•However,thisistoostrictacriteriaforconditionalindependencesincetwonodeswillstillbeconsideredindependentiftheirsimplyexistssomevariablethatdependsonboth.–Forexample,BurglaryandEarthquakeshouldbeconsideredindependentsincetheybothcauseAlarm.11IndependenciesinBayesNets(cont.)•Unlessweknowsomethingaboutacommoneffectoftwo“independentcauses”oradescendentofacommoneffect,thentheycanbeconsideredindependent.–Forexample,ifweknownothingelse,EarthquakeandBurglaryareindependent.•However,ifwehaveinformationaboutacommoneffect(ordescendentthereof)thenthetwo“independent”causesbecomeprobabilisticallylinkedsinceevidenceforonecausecan“explainaway”theother.–Forexample,ifweknowthealarmwentoffthatsomeonecalledaboutthealarm,thenitmakesearthquakeandburglarydependentsinceevidenceforearthquakedecreasesbeliefinburglary.andviceversa.12BayesNetInference•Givenknownvaluesforsomeevidencevariables,determinetheposteriorprobabilityofsomequeryvariables.•Example:GiventhatJohncalls,whatistheprobabilitythatthereisaBurglary?BurglaryEarthquakeAlarmJohnCallsMaryCalls???Johncalls90%ofthetimethereisanAlarmandtheAlarmdetects94%ofBurglariessopeoplegenerallythinkitshouldbefairlyhigh.However,thisignoresthepriorprobabilityofJohncalling.13BayesNetInference•Example:GiventhatJohncalls,whatistheprobabilitythatthereisaBurglary?BurglaryEarthquakeAlarmJohnCallsMaryCalls???Johnalsocalls5%ofthetimewhenthereisnoAlarm.Soover1,000daysweexpect1BurglaryandJohnwillprobablycall.However,hewillalsocallwithafalsereport50timesonaverage.Sothecallisabout50timesmorelikelyafalsereport:P(Burglary|JohnCalls)≈0.02P(B).001AP(J)T.90F.0514BayesNetInference•Example:GiventhatJohncalls,whatistheprobabilitythatthereisaBurglary?BurglaryEarthquakeAlarmJohnCallsMaryCalls???ActualprobabilityofBurglaryis0.016sincethealarmisnotperfect(anEarthquakecouldhavesetitofforitcouldhavegoneoffonitsown).Ontheotherside,eveniftherewasnotanalarmandJohncalledincorrectly,therecouldhavebeenanundetectedBurglaryanyway,butthisisunlikely.P(B).001AP(J)T.90F.0515TypesofInference16SampleInferences•Diagnostic(evidential,abductive):Frome