Advanced Statistical Computing Week 5: EM Algorithm

Transkript

1 Advanced Statistical Computing Week 5: EM Algorithm Aad van der Vaart Fall 2012

2 Contents EM Algorithm Mixtures Hidden Markov models 2

3 EM Algorithm

4 EM-algorithm SETTING: Observation X, likelihood θ p θ (X), hard to maximize and find MLE ˆθ). X can be viewed as 1st coordinate of (X,Y) with density (x,y) p θ (x,y): p θ (x) = p θ (x,y)dµ(y). EM-ALGORITHM: GIVEN θ 0 REPEAT E-step: compute θ E θi ( log pθ (X,Y) X ). M-step: θ i+1 =: point of maximum of this function. θ 0, θ 1,... often tends to MLE, but may not converge, converge slowly, or converge to local maximum. [ Y may be missing data, of augmented data, invented for convenience.] 4

5 EM-Algorithm increases target LEMMA θ 0, θ 1,... generated by EM-algorithm satisfies p θ0 (X) p θ1 (X). PROOF p θ (x,y) = p θ (y x)p θ (x). E θi ( log pθ (X,Y) X ) = E θi ( logpθ (Y X) X ) +logp θ (X). Because θ i+1 maximizes left side over θ, it suffices to show ( E θi logp θi+1 (Y X) X ) ( E θi logp θi (Y X) X ). Or K(p,q):= E p log(q/p)(y) 0 for p =, q =, conditioned on X. p θi p θi+1 Now Kullback-Leibler divergence K(p; q) is nonnegative for any p, q. This does not prove that θ i converges to the MLE! 5

6 EM-Algorithm linear convergence The speed of the EM-algorithm is linear, with slow convergence if the augmented model is statistically much more informative than the data model. 6

7 Mixtures

8 Mixtures SETTING Observations random sample X 1,...,X n from density p θ (x) = k p j f(x;η j ), θ = (p 1,...,p k,η 1,...,η k ). j=1 AUGMENTED DATA P(Y i = j) = p j, X i Y i = j f( ;η j ), i = 1,...,n. Full likelihood p θ (X 1,...,X n,y 1,...,Y n ) = n k ( pj f(x i ;η j ) ) 1 {Yi =j}. i=1j=1 8

9 Mixtures E-step, M-step E-step: given ( p, ẽta): E p, η (log n k ( pj f(x i,η j ) ) ) 1 {Yi =j} X 1,...,X n = i=1j=1 n k log ( p j f(x i,η j ) ) α i,j, i=1j=1 ( ) p j f(x i, η j ) α i,j := P p, η Yi = j X i = c p cf(x i, η c ) = [ k ( n logp j α i,j )]+ j=1 i=1 k [ n logf(x i ;η j ) α i,j ]. j=1 i=1 M-step: for j = 1,...,k: p new j = 1 n n α i,j, i=1 η new j = argmax η n logf(x i ;η) α i,j. i=1 [ If the f( ;η) have a common parameter, then the computation of the η j does not separate as they do here.] 9

10 Mixtures Example EXAMPLE If f( ;η) N(η,1), then n logf(x i,η) α i,j = 1 n 2 i=1 α i,j(x i η) 2 +Const. i=1 η new j = n i=1 α ijx i n i=1 α i,j. EXAMPLE If f( ;η) Γ(r,η), then n logf(x i,η) α i,j = i=1 η new j n (rlogη ηx i ) α i,j +Const. i=1 = r n i=1 α i,j n i=1 α i,jx i. 10

11 R > n=100 > shape=c(2,2,2); eta=c(1,6,.2); prob=c(1/4,1/8,5/8) > component=sample(c(1,2,3),n,replace=true,prob=prob) > x=rgamma(n,shape=shape[component],rate=eta[component]) 11

12 R EM, known shape > k=3; a=matrix(0,n,k); p=c(1/3,1/3,1/3); eta=c(1,2,3); change=1 > while (change>0.0001){ + for (j in 1:k) a[,j]=p[j]*dgamma(x,2,eta[j]) + a=diag(1/apply(a,1,sum))%*%a + etanew=2*apply(a,2,sum)/matrix(x,1,n)%*%a + pnew=apply(a,2,mean) + change=sum(abs(etanew-eta)+abs(pnew-p)) + print(rbind(pnew,etanew)) + eta=etanew; p=pnew} [ --- output deleted ---- ] [,1] [,2] [,3] pnew

13 R packages > library(mixtools) > mod=gammamixem(x,k=3) number of iterations= 323 > summary(mod) Error in summary.mixem(mod) : Unknown mixem object of type gammamixem > mod[[2]]; mod[[3]] [1] comp.1 comp.2 comp.3 alpha beta [ Besides package mixtools, there is also flexmix, and... (?)] 13

14 Mixtures warnings Not all mixtures are identifiable from the data: multiple parameter vectors may give the same mixture. Maximum likelihood may work only if the parameter set is restricted. (Notable example: location scale mixtures, if the scale parameter approaches zero, the likelihood may tend to infinity.) EM tends to be slow for large data sets, and might get stuck in local maxima (?) 14

15 Hidden Markov models

16 Hidden Markov model Y 1 Y 2 Y 3... Y n 1 Y n X 1 X 2 X 3... X n 1 X n Markov chain of hidden states Y 1,Y 2,...,; only outputs X 1,X 2,... observed. X i given Y i conditionally independent of all other variables. EXAMPLES speech recognition: states abstract, outputs Fourier coding of sounds. genomics: states are introns/exons, outputs nucleotides genomics: states are # chromosomal duplicates, outputs noisy measurements genetics: states inheritance vectors, output measured markers. cell biology: states of ion channels, outputs current or no current economics: state of economy, output # firms in default. 16

17 Hidden Markov model Y 1 Y 2 Y 3... Y n 1 Y n X 1 X 2 X 3... X n 1 X n Markov chain of hidden states Y 1,Y 2,...,; only outputs X 1,X 2,... observed. X i given Y i conditionally independent of all other variables. Parameters density π of Y 1 transition density p(y i y i 1 ) of the Markov chain. output density q(x i y i ). Full likelihood π(y 1 )p(y 2 y 1 ) p(y n y n 1 ) q(x 1 y 1 ) q(x n y n ). 17

18 HMM E and M-step E-step: E π, p, q ( logπ(y 1 ) n p(y i Y i 1 ) i=2 n ) q(x i Y i ) X 1,...,X n i=1 = E π, p, q ( logπ(y 1 ) X 1,...,X n ) + + n i=2 n i=1 E π, p, q ( logp(y i Y i 1 ) X 1,...,X n ) E π, p, q ( logq(x i Y i ) X 1,...,X n ). M-step: depends on the specification of models for π,p,q. if state space is finite p is typically left free. only current estimate of law of (Y i 1,Y i ) given X 1,...,X n needed, which are computed using the forward and backward algorithm. 18

19 Baum-Welch The EM-algorithm for the HMM with finite state space, and completely unspecified distributions π, p, q, is called Baum-Welch algorithm. If π and p are left free: π new = p Y 1 X 1,...,X n π, p, q (y). p new (v u) = n i=2 py i 1,Y i X 1,...,X n π, p, q (u, v) n. i=2 py i 1 X 1,...,X n π, p, q (u) If q is also left free (possible for finite output space, but not often the case): q new (x y) = x X i:x i =x py i X 1,...,X i 1,X i =x,x i+1,...,x n π, p, q (y) i:x i =x py i X 1,...,X i 1,X i =x,x i+1,...,x n π, p, q (y). [ To compute these expressions need density of (Y i 1,Y i ) given X 1,...,X n. This is computed using the forward and backward algorithm.] 19

20 Viterbi Y 1 Y 2 Y 3... Y n 1 Y n X 1 X 2 X 3... X n 1 X n The Viterbi algorithm computes the most likely state path given the outcomes: argmax y 1,...,y n P(Y 1 = y 1,...,Y n = y n X 1,...,X n ). 20

21 R > library(hiddenmarkov) > Pi=matrix(c(0.7,0.3,0.2,0.8),2,2,byrow=TRUE); delta=c(0.3,0.7 > n=100; pn=list(size=rep(5,n)); pm=list(prob=c(0.3,0.8)) > myhmm=dthmm(null,pi=pi,delta=delta,distn="binom",pn=pn,pm=pm) > x=simulate(myhmm,n) > > plot(1:n,x$x,type="s",xlab="",ylab="") > lines(1:n,x$y-1,col=2,type="s") [ Markov chain with two states, transition matrix Π, initial distribution δ. Outputs are from the binomial(5,p)- distribution, with θ = 0.3 from state 1 and θ = 0.8 from state 2. Red: states, Black: outputs.] 21

22 R > mod=baumwelch(x); mod$pi; mod$pm [---- output deleted ---] [,1] [,2] [1,] [2,] $prob [1] [ Markov chain with two states, transition matrix Π =, initial distribution δ = (0.3,0.7) Outputs are from the binomial(5,p)- distribution, with θ = 0.3 from state 1 and θ = 0.8 from state 2.] 22

23 R > Viterbi(x) [1] [36] [71] > lines(1:n,viterbi(x)-1,col=3,lw=2) [ Red: true states, Black: outputs; Green: reconstructed states.] 23