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CS639:DataManagementfor
DataScienceLecture14:BayesianMethods
TheodorosRekatsinas1
Announcements
• WewillreleasegradesofMidtermbytheendofdaytoday.
Today– BayesianMethods
• MotivationandIntroduction
• BayesTheorem
• Bayesianinference
Motivation
• Statisticalinference:Drawingconclusionsbasedondatathatissubjecttorandomvariation(observationalerrorsandsamplingvariation)
• Sofarwesawthe“frequentists”pointofview.
• Bayesianinferenceprovidesadifferentwaytodrawconclusionsfromdata.
BasicIdea
• Leveragepriorinformation andupdatepriorinformationwithnewdatatocreateaposteriorprobabilitydistribution.
• Threesteps:• Formprior(aprobabilitymodel)• Conditiononobserveddata(newdatafromyoursample)• Evaluatetheposteriordistribution
BasicIdea
• “The centralfeature ofBayesianinference[is]the directquantificationofuncertainty”(Gelman etal.2014,4).• Lessemphasisonp-valuehypothesistesting.Moreemphasisontheconfidenceandprobabilityintervals.• Manyresearchersactuallyinterpret‘frequentist’confidenceintervals asif theywereBayesianprobabilityintervals.
UncertaintyinFreq.andBayesianApproaches
• Bothinvolvethe estimationofunknownquantities ofinterest• Theestimatestheyproducehave differentinterpretations.
• Frequentist:95%Confidenceinterval:Repeatedsampleswillcontainthetrueparameterwithintheinterval95%ofthetime.
• Bayesian:95%Probability(credible)interval:Thereisa95% probability thattheunknownparameterisactuallyintheinterval.
RandomVariables
ProbabilityDistributions
JointDistributions
MarginalDistributions
ConditionalProbabilities
ConditionalProbabilities
TheProductRule
Bayes’Rule
Bayes’Theorem
Bayes’Theorem
BayesianApproach
BayesianLearning
Wheredopriorscomefrom?
Whendon’tpriormatter(much)?
Whendon’tpriormatter(much)?
Summary