Bayes' Theorem Definition - Investopedia

Bayes' Theorem states that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event ... TableofContents Expand TableofContents WhatIsBayes'Theorem? UnderstandingBayes'Theorem SpecialConsiderations FormulaforBayes'Theorem ExamplesofBayes'Theorem FrequentlyAskedQuestions. WhatIstheHistoryofBayes'Theorem? WhatDoesBayes'TheoremState? WhatIsCalculatedinBayes'Theorem? WhatIsaBayes'TheoremCalculator? HowIsBayes'TheoremUsedinMachineLearning? TheBottomLine CorporateFinance FinancialAnalysis WhatIsBayes'Theorem? Bayes'Theorem, namedafter18th-centuryBritishmathematicianThomasBayes,isamathematicalformulafordeterminingconditionalprobability.Conditionalprobabilityisthelikelihoodofanoutcomeoccurring,basedonapreviousoutcomehavingoccurredinsimilarcircumstances. Bayes'theoremprovidesawaytoreviseexistingpredictionsortheories(updateprobabilities)givenneworadditionalevidence. Infinance,Bayes'Theoremcanbeusedtoratetheriskoflendingmoneytopotentialborrowers.ThetheoremisalsocalledBayes'RuleorBayes'LawandisthefoundationofthefieldofBayesianstatistics. KeyTakeaways Bayes'Theoremallowsyoutoupdatethepredictedprobabilitiesofaneventbyincorporatingnewinformation.Bayes'Theoremwasnamedafter18th-centurymathematicianThomasBayes.Itisoftenemployedinfinanceincalculatingorupdatingriskevaluation.Thetheoremhasbecomeausefulelementintheimplementationofmachinelearning.Thetheoremwasunusedfortwocenturiesbecauseofthehighvolumeofcalculationcapacityrequiredtoexecuteitstransactions. UnderstandingBayes'Theorem ApplicationsofBayes'Theoremarewidespreadandnotlimitedtothefinancialrealm.Forexample,Bayes'theoremcanbeusedtodeterminetheaccuracyofmedicaltestresultsbytakingintoconsiderationhowlikelyanygivenpersonistohaveadiseaseandthegeneralaccuracyofthetest.Bayes'theoremreliesonincorporatingpriorprobabilitydistributionsinordertogenerateposteriorprobabilities. Priorprobability,inBayesianstatisticalinference,istheprobabilityofaneventoccurringbeforenewdataiscollected.Inotherwords,itrepresentsthebestrationalassessmentoftheprobabilityofaparticularoutcomebasedoncurrentknowledgebeforeanexperimentisperformed. Posteriorprobabilityistherevisedprobabilityofaneventoccurringaftertakingintoconsiderationthenewinformation.Posteriorprobabilityiscalculatedbyupdatingthe priorprobability using Bayes'theorem.Instatisticalterms,theposteriorprobabilityistheprobabilityofeventAoccurringgiventhateventBhasoccurred. SpecialConsiderations Bayes'Theoremthusgivestheprobabilityofaneventbasedonnewinformationthatis,ormaybe,relatedtothatevent.Theformulacanalsobeusedtodeterminehowtheprobabilityofaneventoccurringmaybeaffectedbyhypotheticalnewinformation,supposingthenewinformationwillturnouttobetrue. Forinstance,considerdrawingasinglecardfromacompletedeckof52cards. Theprobabilitythatthecardisakingisfourdividedby52,whichequals1/13orapproximately7.69%.Rememberthattherearefourkingsinthedeck.Now,supposeitisrevealedthattheselectedcardisafacecard.Theprobabilitytheselectedcardisaking,givenitisafacecard,isfourdividedby12,orapproximately33.3%,asthereare12facecardsinadeck. FormulaforBayes'Theorem P ( A ∣ B ) = P ( A ⋂ B ) P ( B ) = P ( A ) ⋅ P ( B ∣ A ) P ( B ) where: P ( A ) =  The probability of A occurring P ( B ) =  The probability of B occurring P ( A ∣ B ) = The probability of A given B P ( B ∣ A ) =  The probability of B given A P ( A ⋂ B ) ) =  The probability of both A and B occurring \begin{aligned}&P\left(A|B\right)=\frac{P\left(A\bigcap{B}\right)}{P\left(B\right)}=\frac{P\left(A\right)\cdot{P\left(B|A\right)}}{P\left(B\right)}\\&\textbf{where:}\\&P\left(A\right)=\text{TheprobabilityofAoccurring}\\&P\left(B\right)=\text{TheprobabilityofBoccurring}\\&P\left(A|B\right)=\text{TheprobabilityofAgivenB}\\&P\left(B|A\right)=\text{TheprobabilityofBgivenA}\\&P\left(A\bigcap{B}\right))=\text{TheprobabilityofbothAandBoccurring}\\\end{aligned} ​P(A∣B)=P(B)P(A⋂B)​=P(B)P(A)⋅P(B∣A)​where:P(A)= The probability of A occurringP(B)= The probability of B occurringP(A∣B)=The probability of A given BP(B∣A)= The probability of B given AP(A⋂B))= The probability of both A and B occurring​ ExamplesofBayes'Theorem BelowaretwoexamplesofBayes'TheoreminwhichthefirstexampleshowshowtheformulacanbederivedinastockinvestingexampleusingAmazon.comInc.(AMZN).ThesecondexampleappliesBayes'theoremtopharmaceuticaldrugtesting. DerivingtheBayes'TheoremFormula Bayes'Theoremfollowssimplyfromtheaxiomsofconditionalprobability.Conditionalprobabilityistheprobabilityofaneventgiventhatanothereventoccurred.Forexample,asimpleprobabilityquestionmayask:"Whatistheprobabilityof Amazon.com'sstockpricefalling?"Conditionalprobabilitytakesthisquestionastepfurtherbyasking:"WhatistheprobabilityofAMZNstockpricefallinggiventhattheDowJonesIndustrialAverage(DJIA)indexfellearlier?" TheconditionalprobabilityofAgiventhatBhashappenedcanbeexpressedas: IfAis:"AMZNpricefalls"thenP(AMZN)istheprobabilitythatAMZNfalls;andBis:"DJIAisalreadydown,"andP(DJIA)istheprobabilitythattheDJIAfell;thentheconditionalprobabilityexpressionreadsas"theprobabilitythatAMZNdropsgivenaDJIAdeclineisequaltotheprobabilitythatAMZNpricedeclinesandDJIAdeclinesovertheprobabilityofadecreaseintheDJIAindex. P(AMZN|DJIA)=P(AMZNandDJIA)/P(DJIA) P(AMZNandDJIA)istheprobabilityofboth Aand Boccurring.Thisisalsothesameastheprobabilityof Aoccurringmultipliedbytheprobabilitythat BoccursgiventhatAoccurs,expressedasP(AMZN)xP(DJIA|AMZN).ThefactthatthesetwoexpressionsareequalleadstoBayes'theorem,whichiswrittenas: if,P(AMZNandDJIA)=P(AMZN)xP(DJIA|AMZN)=P(DJIA)xP(AMZN|DJIA) then,P(AMZN|DJIA)=[P(AMZN)xP(DJIA|AMZN)]/P(DJIA). WhereP(AMZN)andP(DJIA)aretheprobabilitiesofAmazonandtheDowJonesfalling,withoutregardtoeachother. TheformulaexplainstherelationshipbetweentheprobabilityofthehypothesisbeforeseeingtheevidencethatP(AMZN),andtheprobabilityofthehypothesisaftergettingtheevidenceP(AMZN|DJIA),givenahypothesisforAmazongivenevidenceintheDow. NumericalExampleofBayes'Theorem Asanumericalexample,imaginethereisadrugtestthatis98%accurate,meaningthat98%ofthetime,itshowsatruepositiveresultforsomeoneusingthedrug,and98%ofthetime,itshowsatruenegativeresultfornonusersofthedrug. Next,assume0.5%ofpeopleusethedrug.Ifapersonselectedatrandomtestspositiveforthedrug,thefollowingcalculationcanbemadetodeterminetheprobabilitythepersonisactuallyauserofthedrug. (0.98x0.005)/[(0.98x0.005)+((1-0.98)x(1-0.005))]=0.0049/(0.0049+0.0199)=19.76% Bayes'Theoremshowsthatevenifapersontestedpositiveinthisscenario,thereisaroughly80%chancethepersondoesnottakethedrug. FrequentlyAskedQuestions. WhatIstheHistoryofBayes'Theorem? ThetheoremwasdiscoveredamongthepapersoftheEnglishPresbyterianministerandmathematician ThomasBayesandpublishedposthumouslybybeingreadtotheRoyalSocietyin1763.LongignoredinfavorofBooleancalculations,Bayes'Theoremhasrecentlybecomemorepopularduetoincreasedcalculationcapacityforperformingitscomplexcalculations.TheseadvanceshaveledtoanincreaseinapplicationsusingBayes'theorem.Itisnowappliedtoawidevarietyofprobabilitycalculations,includingfinancialcalculations,genetics,druguse,anddiseasecontrol. WhatDoesBayes'TheoremState? Bayes'Theoremstatesthat theconditionalprobabilityofanevent,basedontheoccurrenceofanotherevent,isequaltothelikelihoodofthesecondeventgiventhefirsteventmultipliedbytheprobabilityofthefirstevent. WhatIsCalculatedinBayes'Theorem? Bayes'Theoremcalculatestheconditionalprobabilityofanevent,basedonthevaluesofspecificrelatedknownprobabilities. WhatIsaBayes'TheoremCalculator? ABayes’TheoremCalculatorfigurestheprobabilityofanevent A conditionalonanotherevent B,giventhepriorprobabilitiesof A and B,andtheprobabilityof B conditionalon A.Itcalculatesconditionalprobabilitiesbasedonknownprobabilities. HowIsBayes'TheoremUsedinMachineLearning? BayesTheoremprovidesausefulmethodforthinkingabouttherelationshipbetweenadatasetandaprobability.Inotherwords,thetheoremsaysthattheprobabilityofagivenhypothesisbeingtruebasedonspecificobserveddatacanbestatedasfindingtheprobabilityofobservingthedatagiventhehypothesismultipliedbytheprobabilityofthehypothesisbeingtrueregardlessofthedata,dividedbytheprobabilityofobservingthedataregardlessofthehypothesis. TheBottomLine Atitssimplest,Bayes'Theoremtakesatestresultandrelatesittotheconditionalprobabilityofthattestresultgivenotherrelatedevents.Forhighprobabilityfalsepositives,theTheoremgivesamorereasonedlikelihoodofaparticularoutcome. CompareAccounts AdvertiserDisclosure × TheoffersthatappearinthistablearefrompartnershipsfromwhichInvestopediareceivescompensation.Thiscompensationmayimpacthowandwherelistingsappear.Investopediadoesnotincludealloffersavailableinthemarketplace. Provider Name Description RelatedTerms WhatIsPriorProbability? Priorprobability,inBayesianstatisticalinference,istheprobabilityofaneventbasedonestablishedknowledge,beforeempiricaldataiscollected. more ConditionalProbability Conditionalprobabilityisthelikelihoodofaneventoroutcomeoccurringbasedontheoccurrenceofsomeotherpreviouseventoroutcome. more UnderstandingPosteriorProbability Posteriorprobabilityistherevisedprobabilityofaneventoccurringaftertakingintoconsiderationnewinformation. more WhatP-ValueTellsUs P-valueisthelevelofmarginalsignificancewithinastatisticalhypothesistest,representingtheprobabilityoftheoccurrenceofagivenevent. more WhatIsaTwo-TailedTest? Atwo-tailedtestisthestatisticaltestingofwhetheradistributionistwo-sidedandifasampleisgreaterthanorlessthanarangeofvalues. more WhatIsaOne-TailedTest? Aone-tailedtestisastatisticaltestinwhichthecriticalareaofadistributioniseithergreaterorlessthanacertainvalue,butnotboth. more PartnerLinks RelatedArticles Tools HowtoUsetheBayesianMethodofFinancialForecasting MathandStatistics WhatIsaRelativeStandardError? TradingPsychology TheMathBehindBettingOdds&Gambling InvestingEssentials WhattheDowMeansandHowItIsCalculated QuantitativeAnalysis ExplainingTheCapitalAssetPricingModel(CAPM) Separation&Divorce OutlandishDivorceLawsandWhySoManyAreStillontheBooks