Tutkimusmerkitys
Peopleneedtoestimatetheprobabilityofvariousconclusionsinreasoninganddecision-makingbasedonuncertaininformation.Thiskindofreasoningiscalledprobabilisticreasoning.Probabilisticreasoningisnotonlytheresearchobjectofprobabilityandlogic,butalsotheresearchobjectofpsychology,buttheresearchperspectiveisdifferent.Probabilityandlogicstudytheformulasorrulesofobjectiveprobabilityestimation;whilepsychologystudiesthelawsofcognitiveprocessingofpeople'ssubjectiveprobabilityestimation.TheproblemofBayesianreasoningistheproblemofconditionalprobabilityreasoning.Thediscussioninthisfieldhasveryimportanttheoreticalandpracticalsignificanceforrevealingpeople'scognitiveprocessingprocessesandlawsofprobabilityinformation,andguidingpeopletoconducteffectivelearningandjudgmentanddecision-making..
Lauseen määritelmä
Bayesianformula(publishedin1763):
Tämä on kuuluisa "Bayesin lause". Muutamat kirjallisuudet, P(B[1]) ja P(B[2]) kutsutaan perustodennäköisyyksiksi, P(A│B[1]) on hittraatti ja P(A│B[2]) on väärä hälytystaajuus[1].
Sovellusesimerkkejä
Huumausaineiden havaitseminen
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) edustaa todennäköisyyttä, että työntekijät eivät ota huumeita. On selvää, että arvo on 0,995, joka on 1-P(D).
P(+|D) edustaa huumeiden aiheuttajien positiivista havaitsemissuhdetta. Tämä on ehdollinen todennäköisyys ja myös ensisijainen todennäköisyys. Koska positiivisen havaitsemisen tarkkuus on 99%, arvo on 0,99.
P(+|N) edustaa ei-riippuvaisten positiivista havaitsemisprosenttia, eli todennäköisyyttä, että havaitseminen epäonnistuu. Arvo on 0,01, koska ei-riippuvaisille havaitseminen onnegatiivisen todennäköisyydellä 99%, joten todennäköisyys sille, että 99% on virheellinen.
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.
Sijoituspäätös
Bayes'theoremisusedforinvestmentdecisionanalysiswhenthedataofrelatedprojectBisknown,butthereisnodirectdatatoproveprojectA,throughtheanalysisofprojectBThestatusandprobabilityofoccurrenceareanalyzedtoderivethestatusandprobabilityofoccurrenceofAproject.Ifweusemathematicallanguagetodescribe,thatis,whentheprobabilityP(Bi)oftheeventBiisknownandtheprobabilityP(A│Bi)oftheeventAundertheconditionthattheeventBihasoccurred,wecanuseBayes'theoremtocalculatetheoccurrenceoftheeventATheprobabilityoftheeventBiundertheconditionsP(Bi│A).Thebasicstepsforinvestmentdecision-makingaccordingtoBayes'theoremare:
1Luetteloprojektin toteutumistodennäköisyystunnetun projektinB ehdolla, toisin sanoen muuntaa P(A│B)P(B│A);
2Piirrä kaavio;
3Findtheexpectedreturnvalueofeachstatenode,andfilltheresultintothetreediagram;
4Makeinvestmentprojectdecisionsbasedontheanalysisofthetreediagram.
Muut sovellukset
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.