Изследователска значимост
Peopleneedtoestimatetheprobabilityofvariousconclusionsinreasoninganddecision-makingbasedonuncertaininformation.Thiskindofreasoningiscalledprobabilisticreasoning.Probabilisticreasoningisnotonlytheresearchobjectofprobabilityandlogic,butalsotheresearchobjectofpsychology,buttheresearchperspectiveisdifferent.Probabilityandlogicstudytheformulasorrulesofobjectiveprobabilityestimation;whilepsychologystudiesthelawsofcognitiveprocessingofpeople'ssubjectiveprobabilityestimation.TheproblemofБейсianreasoningistheproblemofconditionalprobabilityreasoning.Thediscussioninthisfieldhasveryimportanttheoreticalandpracticalsignificanceforrevealingpeople'scognitiveprocessingprocessesandlawsofprobabilityinformation,andguidingpeopletoconducteffectivelearningandjudgmentanddecision-making..
Дефиниция на теорема
Бейсianformula(publishedin1763):
Това е известната "теорема на Байс". В някои литератури P(B[1]) и P(B[2]) се наричат основни вероятности, P(A│B[1]) е честота на удари и P(A│B[2]) е скорост на фалшива аларма[1].
Примери за приложение
Откриване на drugaddict
Бейс'theoremisveryusefulinthedetectionofdrugaddicts.Assumingthatthesensitivityandreliabilityofaroutinetestresultareboth99%,thatistosay,whenthesubjecttakesdrugs,theprobabilityofeachtestbeingpositive(+)is99%.Whenthesubjectisnottakingdrugs,theprobabilityofeachtestbeingnegative(-)is99%.Judgingfromtheprobabilityofthedetectionresult,thedetectionresultisrelativelyaccurate,butБейс'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) представлява вероятността служителят да не приема наркотици. Очевидно стойността е 0,995, което е 1-P(D).
P(+|D) представлява степента на положително откриване на наркомани. Това е условна вероятност, а също и априорна вероятност. Тъй като точността на положителното откриване е 99%, стойността е 0,99.
P(+|N) представлява степента на положително разпознаване на непристрастени, т.е. вероятността за фалшиво откриване. Стойността е 0,01, тъй като за независимите откриването е Вероятността да бъде отрицателно е 99%, следователно вероятността да бъдете неправилно разпознати като положително е 1-99%.
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%),andthenusingБейс'theoremtocalculate,youwillgetTheprobabilityofthispersontakingdrugsis98.01%.ButthisisnotthestrongestpartofБейс'theorem.IfthispersonisretestedagainandthenrepeatedlycalculatedusingБейс'theorem,theprobabilityofthispersontakingdrugswillbe99.98%(99.9794951%),whichhasexceededthereliabilityofthetest.Spend.
Инвестиционно решение
Бейс'theoremisusedforinvestmentdecisionanalysiswhenthedataofrelatedprojectBisknown,butthereisnodirectdatatoproveprojectA,throughtheanalysisofprojectBThestatusandprobabilityofoccurrenceareanalyzedtoderivethestatusandprobabilityofoccurrenceofAproject.Ifweusemathematicallanguagetodescribe,thatis,whentheprobabilityP(Bi)oftheeventBiisknownandtheprobabilityP(A│Bi)oftheeventAundertheconditionthattheeventBihasoccurred,wecanuseБейс'theoremtocalculatetheoccurrenceoftheeventATheprobabilityoftheeventBiundertheconditionsP(Bi│A).Thebasicstepsforinvestmentdecision-makingaccordingtoБейс'theoremare:
1Избройте вероятността за възникване на проект A при условие на известен проектB, което преобразува P(A│B) в P(B│A);
2Начертайте диаграма на дърво;
3Findtheexpectedreturnvalueofeachstatenode,andfilltheresultintothetreediagram;
4Makeinvestmentprojectdecisionsbasedontheanalysisofthetreediagram.
Други приложения
SearchgiantsGoogleandAutonomy,acompanythatsellsinformationrecoverytools,bothuseБейсianprinciplestoprovidesimilar(buttechnicalTheaboveisnotexact)result.ResearchersalsouseБейсianmodelstodeterminetherelationshipbetweensymptomsanddiseases,createpersonalrobots,anddevelopartificialintelligencedevicesthatcandetermineactionsbasedondataandexperience.
Бейс
Бейс(1701-1761,ThomasБейс),Britishmathematician.BorninLondonin1701,hewasapriest.BecameamemberoftheRoyalSocietyin1742.DiedonApril7,1761.Бейсmainlystudiesprobabilitytheoryinmathematics.Hefirstappliedtheinductivereasoningmethodtothebasictheoryofprobabilitytheory,andfoundedtheБейсianstatisticaltheory,whichmadecontributionstostatisticaldecisionfunctions,statisticalinference,andstatisticalestimation.In1763,RichardPricecollatedandpublishedБейс'result"AnEssaytowardssolvingaProblemintheDoctrineofChances",whichplaysanimportantroleinmodernprobabilitytheoryandmathematicalstatistics.Бейс'otherbook"AnIntroductiontotheDoctrineofOpportunity"waspublishedin1758.ManytermsusedbyБейсianarestillusedtoday.