1. Comp. Genomics
Recitation 6
14/11/06
ML and EM

2. Outline Maximum likelihood estimation HMM Example EM
Baum-Welch algorithm

3. Maximum likelihood One of the methods for parameter estimation
Likelihood: L=P(Data|Parameters)
Simple example:
Simple coin with P(head)=p
10 coin tosses
6 heads, 4 tails
L=P(Data|Params)=(106)p6 (1-p)4

4. Maximum likelihood We want to find p that maximizes L=(106)p6 (1-p)4
Infi 1, Remember?
Log is a monotonically increasing function, we can optimize logL=log[(106)p6 (1-p)4]=
log(106)+6logp+4log(1-p)]
Deriving by p we get: 6/p-4/(1-p)=0
Estimate for p:0.6 (Makes sense?)

5. ML in Profile HMMs Transition Probabilities Emission probabilities
Mi  Mi+1
Mi  Di+1
Mi  Ii
Ii  Mi+1
Ii  Ii
Di  Di+1
Di  Mi+1
Di  Ii
Emission probabilities
Mi  a
Ii  a

6. Parameter Estimation for HMMs
Input: X1,…,Xn independent training sequences
Goal: estimation of  = (A,E) (model parameters)
Note: P(X1,…,Xn | ) = i=1…nP(Xi | ) (indep.)
l(x1,…,xn | ) = log P(X1,…,Xn | ) = i=1…nlog P(Xi | )
Case 1 - Estimation When State Sequence is Known:
Akl = #(occurred kl transitions)
Ek(b) = #(emissions of symbol b that occurred in state k)
Max. Likelihood Estimators:
akl = Akl / l’Akl’
ek(b) = Ek(b) / b’Ek(b’)
small sample, or
prior knowledge correction:
A’kl = Akl + rkl
E’k(b) = Ek(b) + rk(b)

7. Example Suppose we are given the aligned sequences
**---*
AG---C
A-AT-C
AG-AA-
--AAAC AG---C
Suppose also that the “match” positions are marked...

8. Calculating A, E count transitions and emissions: **---* transitions
AG---C
A-AT-C
AG-AA-
--AAAC AG---C
**---*
transitions
emissions

9. Calculating A, E count transitions and emissions: transitions **---*
AG---C
A-AT-C
AG-AA-
--AAAC AG---C
**---*
emissions

10. Estimating Maximum Likelihood probabilities using Fractions
emissions

11. Estimating ML probabilities (contd)
transitions

12. EM - Mixture example
Assume we are given heights of 100 individuals (men/women): y1…y100
We know that:
The men’s heights are normally distributed with (μm,σm)
The women’s heights are normally distributed with (μw,σw)
If we knew the genders – estimation is “easy” (How?)
What we don’t know the genders in our data!
X1…,X100 are unknown
P(w),P(m) are unknown

13. Mixture example
Our goal: estimate the parameters (μm,σm), (μn,σn), p(m)
A classic “estimation with missing data”
(In an HMM: we know the emmissions, but not the states!)
Expectation-Maximization (EM):
Compute the “expected” gender for every sample height
Estimate the parameters using ML
Iterate

14. EM Widely used in machine learning
Using ML for parameter estimation at every iteration promises that the likelihood will consistently improve
Eventually we’ll reach a local minima
A good starting point is important

15. Mixture example
If we have a mixture of M gaussians, each with a probability αi and density θi=(μm,σm)
Likelihood the observations (X):
The “incomplete-data” log-likelihood of the sample x1,…,xN:
Difficult to estimate directly…

16. Mixture example
Now we introduce y1,…,y100: hidden variables telling us what Gaussian every sample came from
If we knew y, the likelihood would be:
Of course, we do not know the ys…
We’ll do EM, starting from θg=(α1g ,..,αMg, μ1g,..,μMg,σ1g,.., σMg)

17. Estimation Given θg, we can estimate the ys! We want to find:
The expectation is over the states of y
Bayes rule: P(X|Y)=P(Y|X)P(X)/P(Y):

18. Estimation
We write down the Q:
Daunting?

19. Estimation
Simplifying:
Now the Q becomes:

20. Maximization Now we want to find parameter estimates, such that:
Infi 2, remember?
To impose the constraint Sum{αi}=1, we introduce Lagrange multiplier λ:
After summing both sides over l:

21. Maximization Estimating μig+1,σig+1 is more difficult 
Out of scope here
What turns out is actually quite straightforward:

22. What you need to know about EM:
When: If we want to estimate model parameters, and some of the data is “missing”
Why: Maximizing likelihood directly is very difficult
How:
Initial guess of the parameters
Finding a proper term for Q(θg, θg+1)
Deriving and finding ML estimators

23. EM estimation in HMMs Input: X1,…,Xn independent training sequences
Baum-Welch alg. (1972):
Expectation:
compute expected # of kl state transitions:
P(i=k, i+1=l | X, ) = [1/P(x)]·fk(i)·akl·el(xi+1)·bl(i+1)
Akl = j[1/P(Xj)] · i fkj(i) · akl · el(xji+1) · blj(i+1)
compute expected # of symbol b appearances in state k
Ek(b) = j[1/P(Xj)] · {i|xji=b} fkj(i) · bkj(i) (ex.)
Maximization:
re-compute new parameters from A, E using max. likelihood.
repeat (1)+(2) until improvement  