PatternRecognitionTheoryandItsApplicationPROBLEMS2.5(1)對C類情況推廣最小錯誤率貝葉斯決策規那么；(2)指出此時使最小錯誤率最小等價于后驗概率最大,即P(i|x)P(j|x)對一切ji成立時,x1.2.5(1)GeneralizetheminimumerrorBayesdecisionruleincaseofclassC;Showthattheminimumerrorrateisequivalenttothemaximumposteriorprobability,namelyP(i|x)P(j|x)wherejiandx1.對兩類問題,證實最小風險貝葉斯決策規那么可表示為假設p(x|J.維__22)p(2),那么x1.p(x|2)(21ii)P(1)'2Inthetwo-categorycase,showthattheminimumriskBayesdecisionrulemaybeexpressedasx1ifp(x|1)￡12——22)p(2).2p(x|2)(2111)p(1)2.7假設11220,1221,證實此時最小最大決策面是來自兩類的錯誤率相等.TOC\o"1-5"\h\z2.7Considerminimaxcriterionfor11220and1221.Provethatinthiscasep〔(error)P2(error).2.22似然比決策準那么為假設l(x)p(x|1)Q-p^^)那么x1P(x|2)p(1)2付對數似然比為h(x)lnl(x),當P(x|i)是均值向量為i和協方差矩陣為i的正態分布時：(1)試推導出h(x),并指出其決策規那么;

(2(2)時,推導h(x)及其決策規那么;(3)分析(1),(2)兩種情況下的決策面類型.2.22Likelihoodratiodecisionrulescanbeexpressedasp(x|1)-p(2)xifl(x)O.p(x|2)p(1)minus-log-likelihoodratiocanbeexpressedash(x)lnl(x),whereP(x|i)~N(i,i).Deduceh(x)andfindthedecisionrule;Let12.Findthedecisionrule;(3)Analyzethedecisionsurfacetypesinquestion(1)andquestion(2).TT2.23二維正態分布,11,0,21,0,12I,p(i)p(2).試寫出負對數似然比決策規那么.2.23LetP(x|i)~N(i,i),11,0T,21,0T,12I,p(1)p(2).Findtheminus-log-likelihoodratiodecisionrule.2.242.24在2.23中,假設那么.,寫出負對數似然比規1Findtheminus-log-likelihoodratiodecisionruleundertheconditionofexercise5習題2.24的情況下,假設考慮損失函數11220,1221,畫出似然比閾與錯誤率之間的關系.⑴求出R(error)0.05時完成Neyman-Pearson決策時總的錯誤率;(2)求出最小最大決策的閾值和總的錯誤率.2.25undertheconditionofexercise3.3let11220,1221.ConsidertheNeyman-Pearsoncriterion,whatistheerrorrateforp(error)0.05;Calculatethethresholdoftheminimaxdecisionandoverallerrorrate.Considerthesampleset=x1,x2,,xNwiththedistributiondensityN,1,-,wherethepriordistributionofispx~N0,1.RespectivelycalculatethemaximumlikelihoodestimateandtheBayesianestimation?.Considerthesampleset=x1,x2,,xNdrawnfromamultivariatenormalpopulationN,2.Respectivelycalculatethemaximumlikelihoodestimate?,?2of,2.Considerthesampleset=x1,x2,,xNdrawnfromabinomialdistributionfx,PPxQ1x,x0,1,0p1,Q1P.CalculatethemaximumlikelihoodestimateP?oftheparameterP.Supposethatthelossfunctionisthequadraticfunction2p?,pp?pandthepriordensityofPfollowstheuniformdistributionfP1,0P1.CalculatetheBayesianestimationP?undertheconditionofexercise3.3.Considerthesampleset=x1,x2,,xNdrawnfromamultivariatenormaldistributionpx~N,whereisknown.Calculatethemaximumlikelihoodestimate?of.

Consideratwo-dimensionallineardiscriminant〔判另fj〕functiongxx2x22〔1〕TransformthediscriminantfunctionintotheformofgxwTx0,anddescribethegeometricfigure〔幾何圖形〕ofgx0;⑵Mapthediscriminantfunctiontoobtainthegeneralized〔廣義〕homogeneous〔齊次〕lineardiscriminantfunctiongxaTy.⑶ShowthattheX-spaceisactuallyasubspaceoftheY-space,andthepartitionoftheX-spacebyaTy0isthesameasthepartitionoftheX-spacebywTx00intheoriginalspace.Describeitbyafigure.8.8.1Giventhreepartitions12,3asshowninthefigurebelow.CalculateS,0.Giventwosamplesets

1Xi0,0,02XiT0,01Xi0,0,02XiT0,0,11X21,0,02X2T0,1,0x31,0,1T0,1,11X4T1,1,02X4T1,1,1CalculatethetransformtoobtainthebiggestJ2expressedbyj2=trs1Sb.9.1Giventwosamplesets1Xi0,0,02XiT0,0,1Xi0,0,02XiT0,0,11X21,0,02X2T0,1,01X31,0,12X3T0,1,11X4T1,1,02X4T1,1,1Respectivelyreducethefeaturespacedimensiontod2andd1,thendescribethepositionsofthesamplesinthefeaturespace.—,410—,410.5LetXi5X2X3andx45,andconsiderthe0followingthreepartitions:⑴1=X1,X