Bayes' Theorem - Definition, Formula, and Example

The Bayes theorem (also known as the Bayes' rule) is a mathematical formula used to determine the conditional probability of events. Home›Resources›Knowledge›Other›Bayes’Theorem Bayes’Theorem Amathematicalformulausedtodeterminetheconditionalprobabilityofevents Writtenby CFITeam UpdatedMay6,2022 WhatistheBayes’Theorem? Instatisticsandprobabilitytheory,theBayes’theorem(alsoknownastheBayes’rule)isamathematicalformulausedtodeterminetheconditionalprobabilityofevents.Essentially,theBayes’theoremdescribestheprobabilityofaneventbasedonpriorknowledgeoftheconditionsthatmightberelevanttotheevent. ThetheoremisnamedafterEnglishstatistician,ThomasBayes,whodiscoveredtheformulain1763.ItisconsideredthefoundationofthespecialstatisticalinferenceapproachcalledtheBayes’inference. Besidesstatistics,theBayes’theoremisalsousedinvariousdisciplines,withmedicineandpharmacologyasthemostnotableexamples.Inaddition,thetheoremiscommonlyemployedindifferentfieldsoffinance.Someoftheapplicationsincludebutarenotlimitedto,modelingtheriskoflendingmoneytoborrowersorforecastingtheprobabilityofthesuccessofaninvestment. Formulafor Bayes’Theorem TheBayes’theoremisexpressedinthefollowingformula: Where: P(A|B)–theprobabilityofeventAoccurring,giveneventBhasoccurred P(B|A)–theprobabilityofeventBoccurring,giveneventAhasoccurred P(A)–theprobabilityofeventA P(B)–theprobabilityofeventB NotethateventsAandBareindependentevents(i.e.,theprobabilityoftheoutcomeofeventAdoesnotdependontheprobabilityoftheoutcomeofeventB). AspecialcaseoftheBayes’theoremiswheneventAisabinaryvariable.Insuchacase,thetheoremisexpressedinthefollowingway: Where: P(B|A–) –theprobabilityofeventBoccurringgiventhatevent A– hasoccurred P(B|A+) –theprobabilityofeventBoccurringgiventhateventA+ hasoccurred Inthespecialcaseabove,eventsA– and A+ aremutuallyexclusiveoutcomesofeventA. Exampleof Bayes’Theorem Imagineyouareafinancialanalystataninvestmentbank.Accordingtoyourresearchofpublicly-tradedcompanies,60%ofthecompaniesthatincreasedtheirsharepricebymorethan5%inthelastthreeyearsreplacedtheirCEOsduringtheperiod. Atthesametime,only35%ofthecompaniesthatdidnotincreasetheirsharepricebymorethan5%inthesameperiodreplacedtheirCEOs.Knowingthattheprobabilitythatthestockpricesgrowbymorethan5%is4%,findtheprobabilitythatthesharesofacompanythatfiresitsCEOwillincreasebymorethan5%. Beforefindingtheprobabilities,youmustfirstdefinethenotationoftheprobabilities. P(A)–theprobabilitythatthestockpriceincreasesby5% P(B)–theprobabilitythattheCEOisreplaced P(A|B)–theprobabilityofthestockpriceincreasesby5%giventhattheCEOhasbeenreplaced P(B|A)–theprobabilityoftheCEOreplacementgiventhestockpricehasincreasedby5%. UsingtheBayes’theorem,wecanfindtherequiredprobability: Thus,theprobabilitythatthesharesofacompanythatreplacesitsCEOwillgrowbymorethan5%is6.67%. RelatedReadings ThankyouforreadingCFI’sguideonBayes’Theorem.Tokeeplearningandadvancingyourcareer,thefollowingresourceswillbehelpful: FreeIntroductiontoStatisticsCourse Forecasting High-LowMethod LawofLargeNumbers NominalData Sharethisarticle GetCertifiedfor
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