中科院机器学习题库-new

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

2、observationsxidrawnfromsuchadistribution:(a)Writedownthelikelihoodasafunctionofthescaleparameterb.(b)Writedownthederivativeoftheloglikelihood.(c)GiveasimpleexpressionfortheMLestimateforb.2、换成Poisson分布：3、二、贝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

3、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,aconjugatepriorisadistributionwithhyperparametersγ,suchthattheposteriordistribution与先验的分布族相同(a)Supposethatthelikelihoodisgivenbytheexpo

4、nentialdistributionwithrateparameterλ:Showthatthegammadistribution_isaconjugatepriorfortheexponential.Derivetheparameterupdategivenobservationsandthepredictiondistribution.(b)Showthatthebetadistributionisaconjugatepriorforthegeometricdistributionwhichdescribesthenumberoftimeacoinisto

5、sseduntilthefirstheadsappears,whentheprobabilityofheadsoneachtossisθ.Derivetheparameterupdateruleandpredictiondistribution.(a)Supposeisaconjugatepriorforthelikelihood;showthatthemixturepriorisalsoconjugateforthesamelikelihood,assumingthemixtureweightswmsumto1.(d)Repeatpart(c)fortheca

6、sewherethepriorisasingledistributionandthelikelihoodisamixture,andthepriorisconjugateforeachmixturecomponentofthelikelihood.somepriorscanbeconjugateforseveraldifferentlikelihoods;forexample,thebetaisconjugatefortheBernoulliandthegeometricdistributionsandthegammaisconjugatefortheexpon

7、entialandforthegammawithfixedα(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，所以极大似然估计的风险最小。（３）回归函数A和

8、B，如果A比B更简单，则