贝叶斯公式的几个应用

[]20080526[](FIB070335-A2-15-C)第27卷第2期大学数学Vol.27,.22011年4月COLLEGEMATHEMATICSApr.2011杨静1,陈冬1,程小红2(1.,100101;2.,100080)[],.[];;[]O211[]C[]16721454(2011)02016604,.,.,.,.,,.1(ThomasBayes,1702-1761),1763,,.1774,(P.s.Laplace,1749-1827).,.,.B1,B2,,Bn,P(Bi)0(i=1,2,,n),AP(A)0,P(BjA)=P(Bj)P(ABj)P(A)(j=1,2,,n),(1),P(A).P(A)=ni=1P(Bi)P(ABi).(2),AB,P(BA)=P(B)P(AB)P(A),(3)P(A)=P(B)P(AB)+P(B)P(AB).(4),B,,P(BA),[1].,.21..,.,.,.,()95%,()99%.,.,.,,,..A={},B={}.,P(B)=0.001,P(AB)=0.95,P(B)=1-0.001=0.999,P(AB)=1-0.99=0.01.(4)P(A)=0.0010.95+0.9990.01=0.01094.(3),P(BA)=0.0010.950.010940.087.,0.087.,.,.,,,,90%,8.7%.,.91.3%.?,,.,.,1000,,10001,999.,10.95+9990.01=10.94,1..,,.,0.0010.087,87.,P(BA)=P(B)P(AB)P(A)=0.0010.050.989060.00006.,,,.:401%.,80%X.,9.6%X.X,?[2],,.,95%70%80%,.B={},A={X}.P(B)=0.01,P(B)=0.99,P(AB)=0.8,P(AB)=0.096.(4),P(A)=P(B)P(AB)+P(B)P(AB)=0.010.8+0.990.096=0.10304.(3),P(BA)=0.010.80.10304=0.0776.167第2期杨静,等:贝叶斯公式的几个应用,X,0.0776.2.?,.T=,L=.,P(TL)=0.88,P(TL)=0.86.,.,.,,0.87.,,,P(L)=0.01,(4),P(T)=P(L)P(TL)+P(L)P(TL)=0.010.88+0.990.14=0.1474.(3),P(LT)=P(L)P(TL)P(T)=0.010.880.14740.06.,94%.,!.,.,P(L)=0.5,P(LT)=0.86,.3..1981330,(JohnHinckleyJr.)..1982,[3].CAT,30%,CAT,2%.CAT,,,..,1.5%.A={CAT};B={}.,P(B)=0.015,P(AB)=0.3,P(B)=1-0.015=0.985,P(AB)=0.02.(4),P(A)=0.0150.3+0.9850.02=0.0242,(3),P(BA)=0.0150.30.0242=0.186.,18.6%,CAT.4..,.,().B={},A={}.P(AB)=0.97,P(AB)=0.95.,6%,.(4),P(A)=P(B)P(AB)+P(B)P(AB)=0.060.97+0.940.05=0.105.168大学数学第27卷(3),P(BA)=0.060.970.105=0.55.,,55%,,..,,.3.,.,,P(ABi),..,,.[][1].[M].:,2001,22.[2]KahnemanD,etal.Judgementunderuncertainty:Heuristicsandbiases[M].Cambridge:CambridgeUniversityPress,1982:249.[3].[J].,2005(10):65-72.SomeApplicationsofBayesianFormulaYANGJing1,CHENDong1,CHENGXiaohong2(1.DepartmentofBasicSubjects,BeijingUnionUniverstiy,Beijing100101,China;2.ElementaryEducationalCollege,CapitalNormalUniversity,Beijing100080,China)Abstract:ThispapergivessomecasesforBayesianformulaandanlysizesthem,whichmighthelpstudentstolearnandunderstandtheformulaanditsapplicationbetter.Keywords:Bayesianformula;theapplication;cases169第2期杨静,等:贝叶斯公式的几个应用