Bayesian Statistics Explained in Simple English For Beginners

“Bayesian statistics is a mathematical procedure that applies probabilities to statistical problems. It provides people the tools to update ... D H M S × Home BayesianStatisticsexplainedtoBeginnersinSimpleEnglish Facebook Twitter Linkedin Youtube NSS— PublishedOnJune20,2016andLastModifiedOnJune11th,2019 Beginner Probability R Statistics Technique Overview ThedrawbacksoffrequentiststatisticsleadtotheneedforBayesianStatistics DiscoverBayesianStatisticsandBayesianInference Therearevariousmethodstotestthesignificanceofthemodellikep-value,confidenceinterval,etc Introduction BayesianStatisticscontinuestoremainincomprehensibleintheignitedmindsofmanyanalysts.Beingamazedbytheincrediblepowerofmachinelearning,alotofushavebecomeunfaithfultostatistics.Ourfocushasnarroweddowntoexploringmachinelearning.Isn’tittrue? Wefailtounderstandthatmachinelearningisnottheonlywaytosolverealworldproblems.Inseveralsituations,itdoesnothelpussolvebusinessproblems,eventhoughthereisdatainvolvedintheseproblems.Tosaytheleast, knowledgeofstatisticswillallowyoutoworkon complexanalyticalproblems,irrespectiveofthesizeofdata. In1770s,ThomasBayesintroduced‘BayesTheorem’.Evenaftercenturieslater,theimportanceof‘BayesianStatistics’hasn’tfadedaway.Infact,todaythistopicisbeingtaughtingreatdepthsinsomeoftheworld’sleadinguniversities. Withthisidea,I’vecreatedthisbeginner’sguideonBayesianStatistics. I’vetriedtoexplaintheconceptsinasimplisticmannerwithexamples.Priorknowledgeofbasicprobability&statistics isdesirable.Youshouldcheckoutthiscoursetogetacomprehensivelowdownonstatisticsandprobability. Bytheendofthisarticle,youwillhaveaconcreteunderstandingofBayesianStatisticsanditsassociatedconcepts.   TableofContents FrequentistStatistics TheInherentFlawsinFrequentistStatistics BayesianStatistics ConditionalProbability BayesTheorem BayesianInference Bernoullilikelihoodfunction PriorBeliefDistribution PosteriorbeliefDistribution TestforSignificance–FrequentistvsBayesian p-value ConfidenceIntervals BayesFactor HighDensityInterval(HDI)   BeforeweactuallydelveinBayesianStatistics,letusspendafewminutesunderstandingFrequentistStatistics,themorepopularversionofstatisticsmostofuscomeacrossandtheinherentproblemsinthat.   1.FrequentistStatistics Thedebatebetweenfrequentistandbayesianhavehauntedbeginnersforcenturies.Therefore,itisimportanttounderstandthedifferencebetweenthetwoandhow doesthereexistsathinlineofdemarcation! It isthemostwidelyusedinferentialtechniqueinthestatisticalworld.Infact,generallyitisthefirstschoolofthoughtthatapersonenteringintothestatisticsworldcomesacross. FrequentistStatistics testswhetheranevent(hypothesis)occursornot.Itcalculates theprobabilityofaneventinthelongrunofthe experiment(i.etheexperimentisrepeatedunderthesameconditionstoobtaintheoutcome). Here,the samplingdistributionsoffixedsizearetaken.Then, theexperimentistheoreticallyrepeatedinfinitenumberoftimesbutpracticallydonewithastoppingintention.Forexample,Iperformanexperimentwithastoppingintentioninmindthat Iwillstoptheexperimentwhenitisrepeated1000timesorIseeminimum300headsinacointoss. Let’sgodeepernow. Now,we’ll understandfrequentiststatistics usinganexampleofcointoss.Theobjectiveistoestimatethefairnessofthecoin.Belowisatablerepresentingthefrequencyofheads: Weknowthatprobabilityofgettingaheadontossingafaircoinis0.5.No.ofheadsrepresentstheactualnumberofheadsobtained.Differenceisthedifferencebetween0.5*(No.oftosses)-no.ofheads. Animportantthingistonotethat,thoughthedifferencebetweentheactualnumberofheadsandexpectednumberofheads(50%ofnumberoftosses)increasesasthenumberoftossesareincreased,theproportionofnumberofheadstototalnumberoftossesapproaches0.5(forafaircoin). Thisexperimentpresentsuswithaverycommonflawfoundinfrequentistapproach i.e. Dependenceoftheresultofanexperimentonthenumberoftimestheexperimentisrepeated. Toknowmoreabout frequentiststatisticalmethods,youcanheadtothisexcellentcourse oninferentialstatistics.   2.TheInherentFlawsinFrequentistStatistics Tillhere,we’veseenjustoneflawinfrequentiststatistics.Well,it’sjustthebeginning. 20thcenturysawamassiveupsurgeinthefrequentiststatisticsbeingappliedtonumericalmodelstocheckwhetheronesampleisdifferentfromtheother,aparameterisimportantenoughtobekeptinthemodelandvariousother manifestationsofhypothesistesting.Butfrequentiststatisticssufferedsomegreatflawsinitsdesignandinterpretation whichposedaseriousconcerninallreallifeproblems.Forexample: 1.p-valuesmeasuredagainstasample(fixedsize)statisticwithsomestoppingintentionchangeswithchangeinintentionandsamplesize.i.eIftwopersonsworkonthesamedataandhavedifferentstoppingintention,theymaygettwodifferent p-values forthesamedata,whichisundesirable. Forexample:PersonAmaychoosetostoptossingacoinwhenthetotalcountreaches100whileBstopsat1000.Fordifferentsamplesizes,weget differentt-scoresanddifferentp-values.Similarly,intentiontostopmaychangefromfixednumberofflipstototaldurationofflipping.Inthiscasetoo,weareboundtogetdifferentp-values. 2-ConfidenceInterval(C.I)likep-valuedependsheavilyonthesamplesize.Thismakesthe stoppingpotentialabsolutelyabsurdsincenomatterhowmanypersonsperformthetestsonthesamedata,theresultsshouldbeconsistent. 3-ConfidenceIntervals(C.I)arenotprobabilitydistributionsthereforetheydonotprovide themostprobablevalueforaparameterandthemostprobablevalues. Thesethreereasonsareenoughtogetyougoingintothinkingaboutthedrawbacksofthefrequentistapproachandwhyisthereaneedforbayesianapproach.Let’sfinditout. Fromhere,we’llfirstunderstandthebasicsofBayesianStatistics.   3.BayesianStatistics “Bayesianstatisticsisamathematicalprocedurethat appliesprobabilitiestostatisticalproblems.Itprovidespeoplethetoolstoupdatetheirbeliefsintheevidenceofnewdata.” Yougotthat?Letmeexplainitwithanexample: Suppose,outofallthe4championshipraces(F1)betweenNikiLaudaandJameshunt,Nikiwon3timeswhileJamesmanagedonly1. So,ifyouweretobetonthewinnerofnextrace,whowouldhebe? IbetyouwouldsayNikiLauda. Here’sthetwist.WhatifyouaretoldthatitrainedoncewhenJameswonandoncewhenNikiwonanditisdefinitethatitwillrainonthenextdate.So,whowouldyoubetyourmoneyonnow? Byintuition,itiseasytoseethatchancesofwinningforJameshaveincreaseddrastically.Butthequestionis:howmuch? Tounderstandtheproblemathand,weneedtobecomefamiliarwithsomeconcepts,firstofwhichisconditionalprobability(explainedbelow). Inaddition,therearecertainpre-requisites: Pre-Requisites: LinearAlgebra:Torefreshyourbasics,youcancheckoutKhan’sAcademyAlgebra. ProbabilityandBasicStatistics: Torefreshyourbasics,youcancheckoutanothercoursebyKhanAcademy.   3.1ConditionalProbability Itisdefinedasthe:ProbabilityofaneventAgivenBequalstheprobabilityofBandAhappeningtogetherdividedbytheprobabilityofB.” Forexample:AssumetwopartiallyintersectingsetsAandBasshownbelow. SetArepresentsonesetofeventsandSetBrepresentsanother.WewishtocalculatetheprobabilityofAgivenBhasalreadyhappened.LetsrepresentthehappeningofeventBbyshadingitwithred. NowsinceBhashappened,thepartwhich nowmattersforAisthepartshadedinbluewhichisinterestingly.So,theprobabilityofAgivenBturnsouttobe: Therefore,wecanwritetheformulaforeventBgivenAhasalreadyoccurredby: or Now,thesecondequationcanberewrittenas: Thisisknownas ConditionalProbability. Let’strytoanswera bettingproblemwiththistechnique. Suppose,BbetheeventofwinningofJamesHunt.Abe theeventofraining.Therefore, P(A)=1/2,sinceitrainedtwiceoutoffourdays. P(B)is1/4,sinceJameswononlyoneraceoutoffour. P(A|B)=1,sinceitrainedeverytimewhenJameswon. Substitutingthevaluesintheconditionalprobabilityformula,wegettheprobabilitytobearound50%,whichisalmostthedoubleof25%whenrainwasnottakenintoaccount(Solveitatyourend). Thisfurtherstrengthenedour belief of Jameswinninginthelightofnewevidencei.erain. Youmustbewonderingthatthisformulabearscloseresemblancetosomethingyoumighthaveheardalotabout.Think! Probably,youguesseditright.ItlookslikeBayesTheorem. Bayes theoremisbuiltontopofconditionalprobabilityandliesintheheart ofBayesianInference.Let’sunderstanditindetailnow.   3.2BayesTheorem BayesTheoremcomesintoeffectwhenmultipleevents formanexhaustivesetwithanothereventB.Thiscouldbeunderstoodwiththehelpofthebelowdiagram.   Now,Bcanbewrittenas So,probabilityofBcanbewrittenas, But So,replacingP(B)intheequationofconditionalprobabilityweget   Thisistheequationof BayesTheorem.   4.BayesianInference Thereisnopointindivingintothetheoreticalaspectofit.So,we’lllearnhowitworks!Let’stakeanexample ofcointossingtounderstandtheideabehindbayesianinference. Animportantpart of bayesianinferenceistheestablishmentofparametersandmodels. Modelsarethemathematicalformulationoftheobservedevents.Parametersarethefactorsinthemodelsaffectingtheobserveddata.Forexample,intossinga coin,fairnessofcoin maybedefinedastheparameterofcoindenotedby θ.TheoutcomeoftheeventsmaybedenotedbyD. Answerthisnow.Whatistheprobabilityof4headsoutof9tosses(D)giventhefairnessofcoin(θ).i.eP(D|θ) Wait,didIasktherightquestion?No. We shouldbe moreinterestedinknowing:Givenanoutcome(D)whatistheprobbailityofcoinbeingfair(θ=0.5) LetsrepresentitusingBayesTheorem: P(θ|D)=(P(D|θ)XP(θ))/P(D) Here,P(θ) istheprior i.ethestrengthofourbeliefinthefairnessofcoinbeforethetoss.Itisperfectlyokaytobelievethatcoincanhaveanydegreeoffairnessbetween0and1. P(D|θ) isthelikelihoodofobservingourresultgivenourdistributionfor θ.Ifweknewthatcoinwasfair,thisgivestheprobabilityofobservingthenumberofheadsinaparticularnumberofflips. P(D)istheevidence.Thisistheprobabilityofdataasdeterminedbysumming(orintegrating)acrossallpossiblevaluesofθ,weightedbyhowstronglywebelieveinthoseparticularvaluesofθ. Ifwehadmultipleviewsofwhatthefairnessofthecoinis(butdidn’tknowforsure),thenthistellsustheprobabilityofseeingacertainsequenceofflipsforallpossibilitiesofourbeliefinthecoin’sfairness. P(θ|D)istheposteriorbeliefofourparametersafterobservingtheevidencei.ethenumberofheads. Fromhere,we’lldivedeeperintomathematicalimplicationsofthisconcept.Don’tworry.Onceyouunderstandthem,gettingtoitsmathematics isprettyeasy. Todefineourmodelcorrectly,weneedtwomathematicalmodelsbeforehand.OnetorepresentthelikelihoodfunctionP(D|θ) andtheotherforrepresentingthedistributionofpriorbeliefs. TheproductofthesetwogivestheposteriorbeliefP(θ|D)distribution. Sincepriorandposteriorarebothbeliefsaboutthedistributionoffairnessofcoin,intuitiontellsusthatbothshouldhavethesamemathematicalform.Keepthisinmind.Wewillcomebacktoitagain. So,thereareseveralfunctionswhichsupporttheexistenceofbayestheorem.Knowingthemisimportant,henceIhaveexplainedthemindetail.   4.1.Bernoulli likelihoodfunction Letsrecapwhatwelearnedaboutthelikelihoodfunction.So,welearnedthat: Itistheprobabilityofobservingaparticularnumberofheadsinaparticularnumberofflipsforagivenfairnessofcoin.Thismeansourprobabilityofobservingheads/tailsdependsuponthefairnessofcoin(θ). P(y=1|θ)=   [Ifcoinisfairθ=0.5,probabilityofobservingheads(y=1)is0.5] P(y=0|θ)= [Ifcoinisfairθ=0.5,probabilityofobservingtails(y=0)is0.5] Itisworthnoticingthatrepresenting1asheadsand0astailsisjustamathematicalnotationtoformulateamodel. Wecancombinetheabovemathematicaldefinitionsintoasingledefinitiontorepresenttheprobabilityofboththeoutcomes. P(y|θ)=  ThisiscalledtheBernoulliLikelihoodFunctionandthetaskofcoinflippingiscalledBernoulli’strials. y={0,1},θ=(0,1) And,whenwewanttoseeaseriesofheadsorflips,itsprobabilityisgivenby: Furthermore,ifweareinterestedintheprobabilityofnumberofheadszturningupinNnumberofflipsthentheprobabilityisgivenby:   4.2.PriorBelief Distribution Thisdistributionisusedtorepresentourstrengthsonbeliefsabouttheparametersbasedonthepreviousexperience. But, whatifonehasnopreviousexperience? Don’tworry.Mathematicianshavedevisedmethodstomitigatethisproblemtoo.Itisknownasuninformativepriors.Iwouldliketoinformyoubeforehandthatitisjustamisnomer.Everyuninformativeprioralwaysprovidessomeinformationeventtheconstantdistributionprior. Well,themathematicalfunctionusedtorepresentthepriorbeliefsisknownas betadistribution. Ithassomeverynicemathematicalpropertieswhichenableustomodelourbeliefsaboutabinomialdistribution. Probabilitydensityfunctionofbetadistributionisoftheform: where,ourfocusstays onnumerator.Thedenominatoristherejusttoensurethatthetotalprobabilitydensityfunctionuponintegrationevaluatesto1. αand βarecalledtheshapedecidingparametersofthedensityfunction.Here αisanalogoustonumberofheadsinthetrialsand βcorrespondstothenumberoftails.Thediagramsbelowwillhelpyouvisualizethebetadistributionsfordifferentvaluesof αand β YoutoocandrawthebetadistributionforyourselfusingthefollowingcodeinR: >library(stats) >par(mfrow=c(3,2)) >x=seq(0,1,by=o.1) >alpha=c(0,2,10,20,50,500) >beta=c(0,2,8,11,27,232) >for(iin1:length(alpha)){     ylibrary(stats) >x=seq(0,1,by=0.1) >alpha=c(13.8,93.8) >beta=c(9.2,29.2) > for(iin1:length(alpha)){    y0.5*(No.oftosses)-no.ofheadsisitcorrect?Reply ciceksays: June20,2016at12:44pm DidyoumisstheindexiofAinthegeneralformulaoftheBayes'theoremonthelefthandsideoftheequation(section3.2)?Reply Bharathsays: June20,2016at2:54pm NicevisualtorepresentBayestheorem,thanksReply AnjaRebbersays: June20,2016at8:01pm Iwillletyouknowtomorrow!BecausetomorrowIhavetodoteachingassistanceinaclassonBayesianstatistics.Iwilltrytoexplainityourway,thenItellyouhowitworkedout. Formeitlooksperfect!Thanks!Reply MelissaLMosersays: June21,2016at6:01am Thoroughandeasytounderstandsynopsis.Goodstuff.Thanks.Reply Gauravsays: June21,2016at9:25am Itwasareallynicearticle,withniceflowtocomparefrequentistvsbayesianapproach.Iwilllookforwardtonextpartofthetutorials.Reply SachinHosamanisays: June21,2016at9:28am Excellentarticle.Ididn'tknewmuchaboutBayesianstatistics,howeverthisarticlehelpedmeimprovemyunderstandingofBayesianstatistics.Reply FangXianfusays: June21,2016at1:31pm Withoutwantingtosuggestthatoneapproachortheotherisbetter,Idon'tthinkthisarticlefulfilleditsobjectiveofcommunicatingin"simpleEnglish". Thecommunicationoftheideaswasfineenough,butifthefocusistobeon"simpleEnglish"thenIthinkthattheterminologyneedstobeintroducedwithmorecare,andmathematicalexplanationsshouldbelimitedandvigorouslyexplained.Reply DavideCaldarasays: June21,2016at4:24pm It'sagoodarticle. AsabeginnerIhaveafewdifficultieswiththelastpart(chapter5)butthepreviouspartswerereallygoodReply Scottsays: June22,2016at1:48am Verynicerefresher.Thankyouandkeepthemcoming.Reply Saileshsays: June22,2016at2:38am Thankyou,NSSforthiswonderfulintroductiontoBayesianstatistics.Thevisualizationswerejustperfecttoestablishtheconceptsdiscussed.AlthoughIlostmywayalittletowardstheend(Bayesianfactor),appreciateyoureffort!Reply Roelsays: June22,2016at9:46am this'stoppingintention'isnotaregularthinginfrequentiststatistics.InfactIonlyhearaboutittoday.Itsortofdistractsmefromthebayesianthingthatistherealtopicofthispost.Perhapsyouneverworkedwithfrequentiststatistics?Reply Roelsays: June22,2016at9:59am Somesmallnotes,butletmemakethisclear:Ithinkbayesianstatisticsmakesoftenmuchmoresense,butIwouldloveitifyouatleastmakethedescriptionofthefrequentiststatisticscorrect.Alsolet'snotmakethisadebateaboutwhichisbetter,it'sasuselessasthepythonvsrdebate,thereisnone. "Inthis,thet-scoreforaparticularsamplefromasamplingdistributionoffixedsizeiscalculated.Then,p-valuesarepredicted.Wecaninterpretpvaluesas(takinganexampleofp-valueas0.02foradistributionofmean100):Thereis2%probabilitythatthesamplewillhavemeanequalto100." Thisisincorrect.ap-valuesayssomethingaboutthepopulation.Youinferenceaboutthepopulationbasedonasample.Ifmean100inthesamplehasp-value0.02thismeanstheprobabilitytoseethisvalueinthepopulationunderthenul-hypothesisis.02.Whichmakesitmorelikelythatyouralternativehypothesisistrue. "samplingdistributionsofdifferentsizes,oneisboundtogetdifferentt-scoreandhencedifferentp-value.Itiscompletelyabsurd." correctitisanestimation,andyoucorrectfortheuncertaintyin Iknowitmakesnosense,wetestforaneffectbylookingattheprobabiltyofascorewhenthereisnoeffect.ifthatisasmallchangewesaythatthealternativeismorelikely.Reply NSSsays: June23,2016at8:04am @Nikhil...Thanksforbringingittothenotice.Itshouldbeno.ofheads-0.5(No.oftosses).Reply NSSsays: June23,2016at8:07am No,Ididn't.:)Reply NSSsays: June23,2016at8:09am ThanksBharath......Itkeepsusmotivated.Reply NSSsays: June23,2016at8:17am @Roel Iagreethispostisn'taboutthedebateonwhichisbetter-BayesianorFrequentist. AndIquoteagain-"Theaimofthisarticlewastogetyouthinkingaboutthedifferenttypeofstatisticalphilosophiesoutthereandhowanysingleofthemcannotbeusedineverysituation". Regardingp-value,whatyousaidiscorrect-Givenyourhypothesis,theprobability.......... Butgenerally,whatpeopleinferis-theprobabilityofyourhypothesis,giventhep-value..... But,stillp-valueisnottherobustmeantovalidatehypothesis,Ifeel. Iwouldliketohearmore. Thanksforcommenting.:)Reply NSSsays: June23,2016at8:23am @Roel Irregularitiesiswhatwecareabout?Isn'tit?andwell,stoppingintentionsdoplayarole.Whatifasasimpleexample:personAperformshypothesistestingforcointossbasedontotalflipsandpersonBbasedontimeduration.Doweexpecttoseethesameresultinboththecases?Reply JoséAvilasays: June23,2016at4:18pm Thisisareallygoodpost!Thanksforsharethisinformationinasimpleway! IhavesomequestionsthatIwouldliketoask! 1)Ididn'tunderstandverywellwhytheC.I."donotprovidethemostprobablevalueforaparameterandthemostprobablevalues".BeforetoreadthispostIwasthinkinginthisway:therealmeanofpopulationisbetweentherangegivenbytheCIwitha,forexample,95%) 2)Ireadarecentpaperwhichstatesthatrejectingthenullhypothesisbybayesfactorat<1/10couldbeequivalentasassumingapvalue<0.001forrejectthenullhypothesis(actually,Idon'trememberverywelltheexactvalues,buttheideaofmakeingthisequivalenceiscorrect?couldbegoodtoapplythisequivalenceinresearch?) 3)Formakingbayesianstatistics,isbettertouseRorPhyton?oritdependsoneachperson?NowImlearningPhytonbecauseIwanttoapplyittomyresearch(Imbiologist!) Thanksinadvanceandsorryformynotsogoodenglish! JoseAvilaReply Shanesays: June24,2016at12:38pm HiNSS, Aquickquestionaboutsection4.2:Ifalpha=no.ofheadsandbeta=no.oftail Whythealphavalue=thenumberoftrailsintheRcode: >alpha=c(0,2,10,20,50,500)#itlookslikethetotalnumberoftrails,insteadofnumberofheads.... >beta=c(0,2,8,11,27,232) Iplottedthegraphsandthesecondonelooksdifferentfromyours... Thanks, ShaneReply AyushMehtasays: June26,2016at9:52am HowcanIknowwhentheotherpostsinthisseriesarereleased?Reply Boomysays: June26,2016at4:36pm ThankyouforthisBlog.IlikeitandIunderstandaboutconceptBayesian.IcanpracticeinRandIcanseesomething. Ithink,youshouldwritethenextguideonBayesianinthenexttime. Iwillwait.Reply Asankasays: June29,2016at8:12am HI... Goodpostandkeepitup...veryuseful...Reply Nikhilsays: August04,2016at7:11pm printerfriendlyversionplease!Reply SharanLobanasays: September21,2016at1:05am Areyousureyouthe'i'inthesubscriptofthefinalequationofsection3.2isn'trequired.IthinkitshouldbeAinsteadofAiontherighthandsidenumerator.Reply Nishthasays: March25,2017at2:13pm HiNSS Thanksforthemuchneededcomprehensivearticle.Pleasetellmeathing:- "SinceHDIisaprobability,the95%HDIgivesthe95%mostcrediblevalues.Itisalsoguaranteedthat95%valueswilllieinthisintervalunlikeC.I." HowisthisunlikeCI?AsfarasIknowCIistheexactsamething.Reply NSSsays: March25,2017at3:45pm @Nishtha....CIistheprobabilityoftheintervalscontainingthepopulationparameteri.e95%CIwouldmean95%ofintervalswouldcontainthepopulationparameterwhereasinHDIitisthepresenceofapopulationparameterinanintervalwith95%probability.Botharedifferentthings.Hopethishelps.Reply Alpersays: June16,2017at6:00pm cicek:ialsothinktheindexiismissinginLHSofthegeneralformulainsubsection3.2(thelastequationinthatsubsection).Reply Stevsays: June19,2017at9:22am Hi,greetingsfromLatam.Ilikedthis.You'vegivenusagoodandsimpleexplanationaboutBayesianStatistics.Helpme,I'venotfoundthenextpartsyet.Reply NSSsays: June28,2017at10:37am Yes,Itisrequired.Ihavemadethenecessarychanges.Reply NSSsays: June28,2017at10:38am Yes,ithasbeenupdated.Thanksforpointingout.Reply LeaveaReplyYouremailaddresswillnotbepublished.Requiredfieldsaremarked*Cancelreply Notifymeoffollow-upcommentsbyemail.Notifymeofnewpostsbyemail.Submit Δ TopResources PythonTutorial:WorkingwithCSVfileforDataScience HarikaBonthu- Aug21,2021 JoinsinPandas:MastertheDifferentTypesofJoinsin.. 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