Význam výzkumu
Peopleneedtoestimatetheprobabilityofvariousconclusionsinreasoninganddecision-makingbasedonuncertaininformation.Thiskindofreasoningiscalledprobabilisticreasoning.Probabilisticreasoningisnotonlytheresearchobjectofprobabilityandlogic,butalsotheresearchobjectofpsychology,buttheresearchperspectiveisdifferent.Probabilityandlogicstudytheformulasorrulesofobjectiveprobabilityestimation;whilepsychologystudiesthelawsofcognitiveprocessingofpeople'ssubjectiveprobabilityestimation.TheproblemofBayesianreasoningistheproblemofconditionalprobabilityreasoning.Thediscussioninthisfieldhasveryimportanttheoreticalandpracticalsignificanceforrevealingpeople'scognitiveprocessingprocessesandlawsofprobabilityinformation,andguidingpeopletoconducteffectivelearningandjudgmentanddecision-making..
Definice věty
Bayesianformula(publishedin1763):
Toto je slavný "Bayesův teorém". V některých literaturách se P(B[1]) a P(B[2]) nazývají základní pravděpodobnosti, P(A│B[1]) je hirát a P(A│B[2]) je rychlost falešného poplachu[1].
Příklady aplikací
Detekce drogových závislostí
Bayes'theoremisveryusefulinthedetectionofdrugaddicts.Assumingthatthesensitivityandreliabilityofaroutinetestresultareboth99%,thatistosay,whenthesubjecttakesdrugs,theprobabilityofeachtestbeingpositive(+)is99%.Whenthesubjectisnottakingdrugs,theprobabilityofeachtestbeingnegative(-)is99%.Judgingfromtheprobabilityofthedetectionresult,thedetectionresultisrelativelyaccurate,butBayes'theoremcanrevealapotentialproblem.Supposeacompanywillconductanopiumusetestforallitsemployees.Itisknownthat0.5%ofitsemployeesusedrugs.Wewanttoknowhowlikelyeachemployeewhohasapositivemedicaltestistotakedrugs.Let"D"beanincidentofdrugusebyemployeesofthecompany,"N"beanincidentwhereemployeesofthecompanydidnottakedrugs,and"+"beanincidentwhereemployeesofthecompanytestedpositive.Available
P(D)representstheprobabilityofanemployeetakingdrugs,regardlessofothercircumstances,thevalueis0.005.Becausethecompany’spre-statisticsindicatethat0.5%ofthecompany’semployeestakedrugs,thisvalueisthepriorprobabilityofD.
P(N) představuje pravděpodobnost, že zaměstnanec nebude brát drogy. Je zřejmé, že hodnota je 0,995, což je 1-P(D).
P(+|D)představuje míru pozitivní detekce drogově závislých. Toto je podmíněná pravděpodobnost a také apriorní pravděpodobnost. Protože přesnost pozitivní detekce je 99 %, je Hodnota 0,99.
P(+|N)představuje míru pozitivní detekce nezávislých, tedy pravděpodobnost falešné detekce. Hodnota je 0,01, protože pro nezávislé je detekce Pravděpodobnost, že budou negativní, je 99%, proto je pravděpodobnost falešné detekce1 - 99 jako pozitivní%.
P(+)representsthepositivedetectionratewithoutconsideringtheinfluenceofotherfactors.Thevalueis0.0149or1.49%.Wecancalculateitbythetotalprobabilityformula:thisprobability=thepositivedetectionrateofdrugusers(0.5%×99%=0.00495)+thepositivedetectionrateofnon-drugusers(99.5%×1%=0.00995).P(+)=0.0149isthepriorprobabilityofapositivetest.Themathematicalformulaisdescribedas :
Accordingtotheabovedescription,wecancalculatesomeoneTheconditionalprobabilityofdrugusewhenthetestispositiveP(D|+):
P(D|+)=P(+|D)P(D)/(P(+|D)P(D)+P(+|N)P(N))=0.99*0.005/0.0149=0.332215
Althoughourtestresultsarehighlyreliable,wecanonlydrawthefollowingconclusions:Ifsomeonetestspositive,thentheprobabilityofthatpersonisdrugtakingisonlyabout33%,whichmeansthatthepersonismorelikelytonottakedrugs.Themoredifficulttheconditionwetested(Dinthiscase,employeedruguse),thegreaterthepossibilityofmisjudgment.
Butifthispersonisre-examinedagain(equivalenttoP(D)=33.2215%,whichistheprobabilityofdrugaddicts,replacingtheoriginal0.5%),andthenusingBayes'theoremtocalculate,youwillgetTheprobabilityofthispersontakingdrugsis98.01%.ButthisisnotthestrongestpartofBayes'theorem.IfthispersonisretestedagainandthenrepeatedlycalculatedusingBayes'theorem,theprobabilityofthispersontakingdrugswillbe99.98%(99.9794951%),whichhasexceededthereliabilityofthetest.Spend.
Investiční rozhodnutí
Bayes'theoremisusedforinvestmentdecisionanalysiswhenthedataofrelatedprojectBisknown,butthereisnodirectdatatoproveprojectA,throughtheanalysisofprojectBThestatusandprobabilityofoccurrenceareanalyzedtoderivethestatusandprobabilityofoccurrenceofAproject.Ifweusemathematicallanguagetodescribe,thatis,whentheprobabilityP(Bi)oftheeventBiisknownandtheprobabilityP(A│Bi)oftheeventAundertheconditionthattheeventBihasoccurred,wecanuseBayes'theoremtocalculatetheoccurrenceoftheeventATheprobabilityoftheeventBiundertheconditionsP(Bi│A).Thebasicstepsforinvestmentdecision-makingaccordingtoBayes'theoremare:
1Uveďte pravděpodobnost výskytuprojektuAzapodmínkyznáméhoprojektuB,to znamená,převeďteP(A│B)naP(B│A);
2Nakreslete stromový diagram;
3Findtheexpectedreturnvalueofeachstatenode,andfilltheresultintothetreediagram;
4Makeinvestmentprojectdecisionsbasedontheanalysisofthetreediagram.
Jiné aplikace
SearchgiantsGoogleandAutonomy,acompanythatsellsinformationrecoverytools,bothuseBayesianprinciplestoprovidesimilar(buttechnicalTheaboveisnotexact)result.ResearchersalsouseBayesianmodelstodeterminetherelationshipbetweensymptomsanddiseases,createpersonalrobots,anddevelopartificialintelligencedevicesthatcandetermineactionsbasedondataandexperience.
Bayes
Bayes(1701-1761,ThomasBayes),Britishmathematician.BorninLondonin1701,hewasapriest.BecameamemberoftheRoyalSocietyin1742.DiedonApril7,1761.Bayesmainlystudiesprobabilitytheoryinmathematics.Hefirstappliedtheinductivereasoningmethodtothebasictheoryofprobabilitytheory,andfoundedtheBayesianstatisticaltheory,whichmadecontributionstostatisticaldecisionfunctions,statisticalinference,andstatisticalestimation.In1763,RichardPricecollatedandpublishedBayes'result"AnEssaytowardssolvingaProblemintheDoctrineofChances",whichplaysanimportantroleinmodernprobabilitytheoryandmathematicalstatistics.Bayes'otherbook"AnIntroductiontotheDoctrineofOpportunity"waspublishedin1758.ManytermsusedbyBayesianarestillusedtoday.