Dirichet Allocation Latent Lecture ( Part 11 2) : Jordan Yuan Andrea Scribes : , , Midterm This Wednesday : Due Sunday Homework 3 : an
Diniohlet Allocation Model Latent Generative : Generative model : Pu Diricwetlw ) on Ad ~ Bh Yan Oa Dirichlet ( x ) K ~ Discrete ( Od ) 2- du~ W Zan Discrete ( Pa ) Na 7dn=h ydnl ~ D
Algorithm Gibbs Sampling Collapsed 1 : fibbs ( topic Updates assignments 1 FT Priori 7dn } { Zdn ( 7 7. 7 \ f. p 1 y ~ du = an , Requirement Implementation Need monginals B to O compute and over . , |dOdB ) , p ,O ) pcy ,7 pcyit = ,wv) Conjugate ( W Dinichlet Bun Od~ Diniohktld , , ... ) ,ak , , ...
Gibbs Collapsed LDA Updates : Fan } { Zal Assignments Update far - Taste \ { yd gaa ~ ~ U Yduiv ) ( L PC Z Odh 7dn= Wwv , Y an an . . , + Nuv =T dn Ndh dn whv ~ - - £du Bv Out = = e- " "+EBv Ni Sufficient Statistics .in?nII7dn=h)Nhv=na.n,=,aIjYdni=YIIZa'nih an q . ] Nail " Nid " Ittaneh ) tdh d 'n'
L( { 0,15 } ,q ) Maximization ( LDA Expectation MAP : Objective log , p* ,9P ) Pc 9 O* Bly ) anymax = = aongygaxlgply 0 , B PLYIQB ) ) ,0,B pczly Lower bound p , ,[ Egan ) ) = log.pl#TjB1sloypiy.0.p ) ) ) Hpczly KL( log ply ,O,p ) qttso , 0,13 = - M E step step - . LIQB [ ( 0 ,p,y ) , 10 ) 4 0,13 = angmaa = argqmax 0 B , 91719,01137 this ) ;¢ ) will qlz ( do =
( Maximization ) Expectation MAP LDA : an P ] Ll{ Egan , ,[ by } , g) = angqngax argampux ;¢|btply # qa ;¢ ) [ log + leg "aYIj÷M ] + leg PIB ) ) Palo log PIG pcyiz ,B ) t angmax = 0 B , Family Representation Exponential , h(x\ [ yt tcx ( acy ] i ) exp ) p × y ) = - )] . ( II #gaµ,§ , B ) ] [ yan=v7I[7an=h ) - leg Bhu Etqcz 17 = ,q , [ { log ) ) ) . ( { Etgn ;¢s[ by 7101 ) Odu Ittaneh Elq , pc = ,
Maximization ( MAP ) Expectation LDA : Lower bound , t.bg#1sbgply.&m B ) , 7. 9. pcy L( { 0,13 } ,q ) Eqa :p = [ { Ndu= ,#17an=h ] ) ¢ndh Expectation Step Fla = : Flpcz , , ,p ,g , [ ) ] ) ) [ I[7an=h Ftq , I[7dn=h ¢ ahh = = , , ,¢ Compute values sufficient statistics of expected Maximization ( exploit ) Step conjugacu : + £I[ + { ydutu ) ¢ Oh cfdnh duh 1 1 wuu - - Odd Phv = = - - + £ + fnI[ { ]¢dnl [ l cfdnl de yaiu - 1 . weu e
LDA Algorithms Other Sampling : Monte Carlo Sequential i ) Requirement Sequence densities intermediate of : ) Marginal PC 994in 7 di :n Single , document p 0 : over B , tin ) ( yn Zd B ) Pcyd / 7dam = iin , , , Marginal aw0 ← 9( I Fd a ) I For 7A 7dm yd lin ,n = PC - tin - , , , :n , - , , " ( Analogous Gibbs sampling ) to Compute was Example Question the weights importance : for generations h > 1
) Sequential ( Marte Carlo General Formulation Assume Unnmmalized Densities jfkt ) , ) 1 × g. : ... . Importance Sampling First step : s , ) ws ycxsilqkil 9 ( × x. ~ :-. , Propose Subsequent steps from samples previous : ( win int Discrete at ' ~ } ' " ins , . , ' !×" , ) x. ~ it + , . . xst~qcxi.ly?I.7x!t:=x!.x.aiiIi ' , ) xt qcxtix it ~ , . 8+45 .sn#sqcxilxa?I vi. ) it ' Incremental weiswl is a , ' . ,K ft
LDA Other Algorithms Sampling : pPlYd Carlo Monte Sequential .nl?amil3)p(7d,nlbd,i:n-i,7d.l:n , ) i - :^) ( ) yn Fd ,7d pcyd = ' in ' 1 , , , i ,&d ) 9( I Fd For ( P ) tap 7d,i Yd P , iin 1 lin ,n = i - :n i - - , , , .nl/B)pCyd.l:nu,7d.l:n.i Faint Jul 7h :n ) Wns bdy.to?In...lPCyd.i:n%7ad?iin.i1pcydn:n.,#iInilp1pi7'a,nl7IiYnYyd.iin.i.p , = , are ah . Ku , ( ) , ,7d ) 7d 91 Yd :ni , ,i:n , ;n i , . - , pcgda -17in PHI 17in Plbdn .nl ,B > = ) = µMpwI[9nn=D ) Isaiah , B) = v
LDA Algorithms Other Sampling : Hamiltonian Carlo Monte Requires log Gradient of joint : - Vqp log Tqp UC 0,13 ) R ) pcy ,O = , to L ( to .pl ,¢ ) ¢dnEe [ Ittndneh ) ) Epa ,D,p ) iy Example Question LDA Suppose to : were run you Wikipedia of all Would recommend you using on . HMC to ) ? 0,13 plop sample ly x
Monte Carlo Hamiltonian Algorithm : %# =Is ' X. ' step ) ( Single finlpl HMC • . • 2 :X !£5 th ) Norm 18 Fi ~ , x. Edit • , Is ' ' ÷ . e. It ,p , , ,p , ,M,E ,T ) - ← . ( RU , I LEAPFROG ;= - Ft ) ] ) min ( . HKF expl a l = , , How times expttlki many , -511 ) , Tqp UIO .pl ? Uniform ( 0,1 ) calling u are we ~ It T £4 times " { Is ,= Is " d > u
LDA Algorithms Other Sampling : Example Question LDA to Suppose : were run you Wikipedia of all Would recommend you using on . HMC to ) ? 0,13 PIQP sample ly x Answer No for Computing the each gradient ; . leapfrog would full step the data require a press over .
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