Researchsignificance
Peopleneedtoestimatetheprobabilityofvariousconclusionsinreasoninganddecision-makingbasedonuncertaininformation.Thiskindofreasoningiscalledprobabilisticreasoning.Probabilisticreasoningisnotonlytheresearchobjectofprobabilityandlogic,butalsotheresearchobjectofpsychology,buttheresearchperspectiveisdifferent.Probabilityandlogicstudytheformulasorrulesofobjectiveprobabilityestimation;whilepsychologystudiesthelawsofcognitiveprocessingofpeople'ssubjectiveprobabilityestimation.TheproblemofBayesianreasoningistheproblemofconditionalprobabilityreasoning.Thediscussioninthisfieldhasveryimportanttheoreticalandpracticalsignificanceforrevealingpeople'scognitiveprocessingprocessesandlawsofprobabilityinformation,andguidingpeopletoconducteffectivelearningandjudgmentanddecision-making..
Theoremdefinition
Bayesianformula(publishedin1763):
ThisItisthefamous"Bayes'Theorem".Insomeliteratures,P(B[1])andP(B[2])arecalledbasicprobabilities,P(A│B[1])isthehitrate,andP(A│B[2])isthefalsealarmrate[1].
Applicationexamples
Drugaddictdetection
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)representstheprobabilityofanemployeenottakingdrugs.Obviously,thevalueis0.995,whichis1-P(D).
P(+|D)representsthepositivedetectionrateofdrugaddicts.Thisisaconditionalprobabilityandalsoaprioriprobability.Sincethepositivedetectionaccuracyis99%,theThevalueis0.99.
P(+|N)representsthepositivedetectionrateofnon-addicts,thatis,theprobabilityoffalsedetection.Thevalueis0.01,becausefornon-addicts,thedetectionisTheprobabilityofbeingnegativeis99%,therefore,theprobabilityofbeingfalselydetectedaspositiveis1-99%.
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.
Investmentdecision
Bayes'theoremisusedforinvestmentdecisionanalysiswhenthedataofrelatedprojectBisknown,butthereisnodirectdatatoproveprojectA,throughtheanalysisofprojectBThestatusandprobabilityofoccurrenceareanalyzedtoderivethestatusandprobabilityofoccurrenceofAproject.Ifweusemathematicallanguagetodescribe,thatis,whentheprobabilityP(Bi)oftheeventBiisknownandtheprobabilityP(A│Bi)oftheeventAundertheconditionthattheeventBihasoccurred,wecanuseBayes'theoremtocalculatetheoccurrenceoftheeventATheprobabilityoftheeventBiundertheconditionsP(Bi│A).Thebasicstepsforinvestmentdecision-makingaccordingtoBayes'theoremare:
1ListtheoccurrenceprobabilityofprojectAundertheconditionofknownprojectB,thatis,convertP(A│B)toP(B│A);
2Drawatreediagram;
3Findtheexpectedreturnvalueofeachstatenode,andfilltheresultintothetreediagram;
4Makeinvestmentprojectdecisionsbasedontheanalysisofthetreediagram.
Otherapplications
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.