01N1呪(X")=—万+臣匸2、换成 Poisson 分布：p(xIe,y=0,1,2,…x!ii1-i=1机器学习题库、极大似然1、MLestimationofexponentialmodel(10)AGaussiandistributionisoftenusedtomodeldataontherealline,butissometimesinappropriatewhenthedataareoftenclosetozerobutconstrainedtobenonnegative.Insuchcasesonecanfitanexponentialdistribution,whoseprobabilitydensityfunctionisgivenbyp(x)=GivenNobservationsxidrawnfromsuchadistribution:(a) Writedownthelikelihoodasafunctionofthescaleparameterb.(b) Writedownthederivativeoftheloglikelihood.(c) GiveasimpleexpressionfortheMLestimateforb.N.,(a)Z(X:b)=口产-琴=Z>-NeY 匸=i 处i-1°l(X:b)=log(Z(X:6))=-Nlog(b)-+刀08=1N»=11N—(X:t)=0=>fc=-^xi=i=ll(&)=区 log(p(xIB))=区 xlog0-0-log(x!)iii=1log0-N0-区 log(x!)ii=1二、贝叶斯1、贝叶斯公式应用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 im，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。：()p(known,correct)ppxknownIcorrect)==—p(known)p+(1-p)丄m2、ConjugatepriorsGivenalikelihoodp(x|g)foraclassmodelswithparameters0,aconjugatepriorisadistributionp(g|丫)withhyperparametersY,suchthattheposteriordistributionp(0|X,丫)=ap(X10)p(0|y)=p(0|y,)与先验的分布族相同(a) Supposethatthelikelihoodisgivenbytheexponentialdistributionwithrateparameter 入：[P(A|X)x沪 Nexp(—0入)[[入 exp(—入:右)%)匕3aC『(异"5(〉(05))gamma(A\a+N,0+SN).(0+ SN 严 『(a+N)(0+SJV+畑+1户+N,(0+SN)a+Na+NF(Q+N)Xgamma(入|a+N：0+SN+ZN+i)dp(xI 九)=九 e-九 xShowthatthegammadistributionGamma(入 Ia,卩)=_入 a-ie-3 九一isaconjugatepriorfortheexponential.Derivetheparameterupdategivenobservationsx...xand1””Nthepredictiondistributionp(x|x...x).N+11N(a) ExponentialandGammaThelikelihoodisP(X|A)二 nil入 exp(—入工运)andthepriorisp(X\a./3)=gamma(入|Q：0)=exp(—0入).LetXdenotetheobservations巾,…必 NandletSNdenotetheirsum.ThentheposteriorisThereforetheparameterupdatesareasfollows:Forthepredictiondistributionwecomputethefollowingintegr...
