中科院机器学习题库-new.doc

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1、....机器学习题库一、极大似然1、MLestimationofexponentialmodel(10)AGaussiandistributionisoftenusedtomodeldataontherealline,butissometimesinappropriatewhenthedataareoftenclosetozerobutconstrainedtobenonnegative.Insuchcasesonecanfitanexponentialdistribution,whoseprob

2、abilitydensityfunctionisgivenbyGivenNobservationsxidrawnfromsuchadistribution:(a)Writedownthelikelihoodasafunctionofthescaleparameterb.(b)Writedownthederivativeoftheloglikelihood.(c)GiveasimpleexpressionfortheMLestimateforb.2、换成Poisson分布：........3、一、贝

3、叶斯假设在考试的多项选择中，考生知道正确答案的概率为p，猜测答案的概率为1-p，并且假设考生知道正确答案答对题的概率为1，猜中正确答案的概率为，其中m为多选项的数目。那么已知考生答对题目，求他知道正确答案的概率。1、ConjugatepriorsThereadingsforthisweekincludediscussionofconjugatepriors.Givenalikelihoodforaclassmodelswithparametersθ,aconjugatepriorisadistri

4、butionwithhyperparametersγ,suchthattheposteriordistribution与先验的分布族相同(a)Supposethatthelikelihoodisgivenbytheexponentialdistributionwithrateparameterλ:Showthatthegammadistribution_isaconjugatepriorfortheexponential.Derivetheparameterupdategivenobservati

5、onsandthepredictiondistribution.........(a)Showthatthebetadistributionisaconjugatepriorforthegeometricdistributionwhichdescribesthenumberoftimeacoinistosseduntilthefirstheadsappears,whentheprobabilityofheadsoneachtossisθ.Derivetheparameterupdaterulean

6、dpredictiondistribution.........(a)Supposeisaconjugatepriorforthelikelihood;showthatthemixturepriorisalsoconjugateforthesamelikelihood,assumingthemixtureweightswmsumto1.(d)Repeatpart(c)forthecasewherethepriorisasingledistributionandthelikelihoodisamix

7、ture,andthepriorisconjugateforeachmixturecomponentofthelikelihood.somepriorscanbeconjugateforseveraldifferentlikelihoods;forexample,thebetaisconjugatefortheBernoulliandthegeometricdistributionsandthegammaisconjugatefortheexponentialandforthegammawith.

8、.......fixedα(e)(Extracredit,20)Explorethecasewherethelikelihoodisamixturewithfixedcomponentsandunknownweights;i.e.,theweightsaretheparameterstobelearned.三、判断题（1）给定n个数据点，如果其中一半用于训练，另一半用于测试，则训练误差和测试误差之间的差别会随着n的增加而减小。（2）极大似然估计是无偏估计且在所有的无偏估计中方差最小