1[Aug 27]Tutorial [Ariel Kleiner].pdf

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1、CS294-34:PracticalMachineLearningTutorialArielKleinerContentinspiredbyFall2006tutoriallecturebyAlexandreBouchard-CoteandAlexSimmaAugust27,2009MachineLearningDrawsHeavilyOn...ProbabilityandStatisticsOptimizationAlgorithmsandDataStructuresProbability:FoundationsAprobabilityspace(;F;P)co

2、nsistsofasetof"possibleoutcomes"aset1Fofevents,whicharesubsetsofaprobabilitymeasureP:F![0;1]whichassignsprobabilitiestoeventsinFExample:RollingaDieConsiderrollingafairsix-sideddie.Inthiscase,=f1;2;3;4;5;6gF=f;;f1g;f2g;:::;f1;2g;f1;3g;:::g11P(;)=0;P(f1g)=;P(f3;6g)=;:::631Actually,Fisa

3、-field.SeeDurrett’sProbability:TheoryandExamplesforthoroughcoverageofthemeasure-theoreticbasisforprobabilitytheory.Probability:RandomVariablesArandomvariableisanassignmentof(oftennumeric)valuestooutcomesin.ForasetAintherangeofarandomvariableX,theinducedprobabilitythatXfallsinAiswritten

4、asP(X2A).ExampleContinued:RollingaDieSupposethatwebet$5thatourdierollwillyielda2.LetX:f1;2;3;4;5;6g!f5;5gbearandomvariabledenotingourwinnings:X=5ifthedieshows2,andX=5ifnot.Furthermore,15P(X2f5g)=andP(X2f5g)=:66Probability:CommonDiscreteDistributionsCommondiscretedistributionsforara

5、ndomvariableX:Bernoulli(p):p2[0;1];X2f0;1gP(X=1)=p;P(X=0)=1pBinomial(p;n):p2[0;1];n2N;X2f0;:::;ngnxnxP(X=x)=p(1p)xThemultinomialdistributiongeneralizestheBernoulliandtheBinomialbeyondbinaryoutcomesforindividualexperiments.Poisson():2(0;1);X2NexP(X=x)=x!Probability:MoreonRand

6、omVariablesNotation:XPmeans"XhasthedistributiongivenbyP"Thecumulativedistributionfunction(cdf)ofarandomvariableX2Rmisdefinedforx2RmasF(x)=P(Xx).WesaythatXhasadensityfunctionpifwecanwriteRxP(Xx)=p(y)dy.1Inpractice,thecontinuousrandomvariableswithwhichwewillworkwillhavedensities.Forc

7、onvenience,intheremainderofthislecturewewillassumethatallrandomvariablestakevaluesinsomecountablenumericset,R,orarealvectorspace.Probability:CommonContinuousDistributionsCommoncontinuousdistributionsforarandomvariableX:Uniform(a;b):a;b2R,a

8、1(x