Pattern Recognition and Machine Learning

572 11.CONTINUOUSLAT!::NTVANIM1LI::S

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Flgu..12.9 I\n~I"'tfat""oIlt>eII"""fativevi&woI1t>epabi!;st",PeAmodeIfof"two-dimensiooal<!ala
spaceandaon&-<lirnent.ionallat/l<1tspace, AnOb&erved<!alapointxIsgeneratedbyfirstdrawinga valuei
fof1t>eIat&n1vafiatlle/f(lm~spriordist,~t""P(~)andItlendrawinga val""fofx lromaniSO/fopK:Gaussian
distr~t""(iijust,al&(lbytheredcir<:ie's)havingmeanwi+"andcoY8r1.once,,'1 Thel/f&er\ellips.&$showl!le
density"""toors!ofthemarg'''''1dis1r1bulionPIx).

lossofge""raJityinassumingazeromean.unitco\'arianceGau"ianforthelatent

distributi"nII{Z)becausea moregcneralGau"i3ndi"ributi"nwouldgi"erisetoan

equivalentprobabili"icn>odel.

WecanviewtheprobabilisticPeAmodelfroma geoerati"e\'iew""intin"hich

a sampled'-alueoftheob""Yed,..riable isobIainedbyfirstchoo,inga ,..Iuefor

thelatent,'ariahleaodthen>amplingtheOO",,,'e;j,-ariablecooditionedonthislao

tent\'alue, Specifically,theV-dimen'ionalOO"''''ed'-ariablex is definedbya lin·

ea,tran,formati,,"ofthe'\/·dimen,i"nallatcnt'-ariablez plu,additi'-eGaussian

'noise',<0that

,,=\VZ+,,+~ (12.33)

w!>erez isan M-di""'nsionalGaussianlalentvariable.and..isa V·dimensi"nal

,ero-meanGau..ian-distributednoi..,"ariablewitbco'-ariance,,21.Thisgenerative

processis illustratedinFigure12.9.NOIethatthisframe".-orl<isbasedona mapping

fromlatent,pace 10 dataspace.incontrast 10 thenl()l'(:C(""'cnti,,,,"1"iew"fI'CA

dis.cus"'dalx",e,11Ie",,'e=mapping,fromdataspacetothelatentspace.,,-illhe

OOlained,honlyusingHaycs·lhwn:m.

SUf!ll'OSCwewish 10 deten"inethe"aluesofll>oparameters\V.I'and,,'uSIng

maximumlikelihuo<l,Towrite""""nlhelikeliltoodfunction,weneedan""pression

fortl>omarginaldistributioop{")oftl>o~,,'ed...riahle_Thisisexprt__sed.fmn'

thesumaodp,oductrules"fprobability,intheform

(11,34)

E,e,,-ise 12,7

ll""auS(:thiscorrespondstoalinear·Gau"i,nlT1(llIcLthi<marginaldi,tribulionis

againGaussian.atldis givenby

,,(,,)_N{xllf,C) (IUS)