Pattern Recognition and Machine Learning
-, o , -, o , -, o , o , ", • , , ,<, , (^0) ~, 0 0 .... -, -, -, , 0 , -, 0 -, 0 , , ", , , -, Flgu..12.12 Synt"'elic& ...
582 12.CONTINUOUSLATENTVARIABLES Figure12.13 ProbabilisticgraphicalmodelforBayesianpeAin whichthedistributionovertheparametermat ...
Section4.4 Section3.5.3 12.2.ProbabilisticpeA 583 Becausethisintegrationisintractable,wemakeuseoftheLaplaceapproxima- tion.Ifwea ...
584 12.CONTINUOUSLATENTVARIABLES • •• • • • • • • •• • • • • • •• • • • • • • • • • • •• • • Figure12.14 ...
12.2.l'ru":ohilislkI'CA 585 FllIure12.15 Gillbs.,,,,>p!j"lllo<Bay<lslan PCAsh<Mingplots oj Ino, versus ~eralion numb ...
586 12.CONTINUOUSLATENTVARIABLES Exercise 12.22 tocomputeinO(D)steps),whichis convenientbecauseoftenM « D.Similarly, theM-step e ...
12.3.K~mcil'Co'. 587 ...'\ (12.73) Figu'.12.16 SctIematic_,.lion 01 kernelPeA.A<UtllHIInlheOflglnal<Utaspacel~'_plot}.. Pf ...
588 12.CONTINUOUSLATENTVARIABLES Substitutingthisexpansionbackintotheeigenvectorequation,weobtain 1 N N N N L ¢(xn)¢(xn)TL aim¢( ...
12.3.KernelPCA 589 Sofarwehaveassumedthattheprojecteddatasetgivenby¢(xn)haszero mean,whichingeneralwillnotbethecase. Wecannotsim ...
590 12.CONTINUOUSLATENTVAlUABLES _. (12.88) Figure12.11 E"llmple 01 kernelPCA,withaGaussiankernelawIiOO 10 asynthetic<latasat ...
12.4.NonlinearLatentVariableModels 591 12.4 Nonlinear Latent Variable Models Exercise 12.28 Inthischapter,wehavefocussedonthesim ...
592 12.CONTINUOUSLATENTVARIABLES Exercise 12.29 likelihoodfunctionforthismodelis a functionofthecoefficientsinthelinearcom- bina ...
12.4.NonlinearLatentVariableModels 593 Figure12.18 AnautoassociativemUltilayerperceptronhaving twolayersof weights.Sucha network ...
594 12.CONTINUOUSLATENTVARIABLES Figure12.19 Addition of extra hidden lay- ersofnoolinearunitsgivesan auloassocialivenetworkwhic ...
12.4.NonlinearLatentVariableModels 595 Ithastheadvantageofnotbeinglimitedtolineartransformations, althoughit con- tainsstandardp ...
596 12.CONTINUOUSLATENTVARIABLES >..= gf(X)becauseit dependsontheparticularcurvef(>"). Fora continuousdata densityp(x),a p ...
Chapter 5 ChapterJJ 12.4.NonlinearLatentVariableModels 597 fold. Forinstance,if twopointslieona circle,thenthegeodesicisthearc-l ...
598 12.CONTINUOUSLATENTVAK1AHU':S .•.'" FllIu.e12.21 ?lotottrleoillkYw<:lataWllisualiz.edusingPeAontheleftandGTMonItlengr,tFO ...
Exercises 599 bilisticfoundationalsomakesit verystraightforwardtodefinegeneralizationsof GTM(Bishopet al.,1998a)suchasa Bayesian ...
600 12.CONTINUOUSLATENTVARIABLES theeigenvectorsofS.Becausethesesolutionsareallequivalent,itisconvenientto choosetheeigenvectors ...
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