parameter estimation for hmm lecture #7

.

Parameter Estimation For HMM Lecture #7

Background Readings: Chapter 3.3 in the text book, Biological Sequence Analysis, Durbin et al., 2001.

2

Reminder: Hidden Markov Model

Markov Chain transition probabilities: p(Si+1= t|Si = s) = ast

Emission probabilities: p(Xi = b| Si = s) = es(b)

S1 S2 SL-1 SL

x1 x2 XL-1 xL

M M M M

TTTT

11 11

( , ) ( , , ; ,..., ) ( )i i i

L

L L s s s ii

p p s s x x a e xs x

For each integer L>0, this defines a probability space in which the simple events are HMM of length L.

3

Hidden Markov Model for CpG Islands

The states: Domain(Si)={+, -} {A,C,T,G} (8 values)

In this representation P(xi| si) = 0 or 1 depending on whether xi is consistent with si . E.g. xi= G is consistent with si=(+,G) and with si=(-,G) but not with any other state of si.

A- T+

A T

G +

G

… …… …

4

Reminder: Most Probable state pathS1 S2 SL-1 SL

x1 x2 XL-1 xL

M M M M

TTTT

Given an output sequence x = (x1,…,xL),

A most probable path s*= (s*1,…,s*

L) is one which maximizes p(s|x).

1( ,..., )

* *1 1 1* ( ,..., ) ( ,..., | ,..., )argmax

Ls s

L L Ls s s p s s x x

6

A- C- T- T+

A C T T

G +

G

Predicting CpG islands via most probable path:

Output symbols: A, C, G, T (4 letters). Markov Chain states: 4 “-” states and 4 “+” states, two for each letter (8 states total).The most probable path (found by Viterbi’s algorithm) predicts CpG islands.

Experiment (Durbin et al, p. 60-61) shows that the predicted islands are shorter than the assumed ones. In addition quite a few “false negatives” are found.

7

Reminder: Most probable state

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

Given an output sequence x = (x1,…,xL), si is a most probable state (at location i) if:

si=argmaxk p(Si=k |x).

),..,|(),..,(

),..,|(),..,|( Lii

L

LiiLii xxsSp

xxp

xxsSpxxsSp 1

1

11

9

Finding the probability that a letter is in a CpG island via the algorithm for most probable state:

The probability that the occurrence of G in the i-th location is in a CpG island (+ state) is:

∑s+ p(Si =s+ |x) = ∑s+ F(Si=s+ )B(Si=s+

)

Where the summation is formally over the 4 “+” states, but actually

only state G+ need to be considered (why?)

A- C- T- T+

A C T T

G +

G

i

10

Parameter Estimation for HMM

An HMM model is defined by the parameters: akl and ek(b), for all states k,l and all symbols b. Let θ denote the collection of these parameters.

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

lk

b

akl

ek(b)

11

Parameter Estimation for HMM

To determine the values of (the parameters in) θ, use a training set = {x1,...,xn}, where each xj is a sequence which is assumed to fit the model.Given the parameters θ, each sequence xj has an assigned probability p(xj|θ).

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

12

Maximum Likelihood Parameter Estimation for HMM

The elements of the training set {x1,...,xn}, are assumed to be independent, p(x1,..., xn|θ) = ∏j p (xj|θ).

ML parameter estimation looks for θ which maximizes the above.

The exact method for finding or approximating this θ depends on the nature of the training set used.

13

Data for HMM

Possible properties of (the sequences in) the training set:1.For each xj, the information on the states sj

i (the symbols

xji are usually known).

2.The size (number of sequences) of the training set

S1 S2 SL-1 SL

x1 x2 XL-1 xL

M M M M

TTTT

14

Case 1: ML when Sequences are fully known

We know the complete structure of each sequence in the training set {x1,...,xn}. We wish to estimate akl and ek(b) for all pairs of states k, l and symbols b.

By the ML method, we look for parameters θ* which maximize the probability of the sample set: p(x1,...,xn| θ*) =MAXθ p(x1,...,xn| θ).

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

15

Case 1: Sequences are fully known

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

Let mkl= |{i: si-1=k,si=l}| (in xj). mk(b)=|{i:si=k,xi=b}| (in xj).

( )

( , ) ( , )

then: ( | ) [ ( )]kl km m bkkl

k l k b

jp a e bx

For each xj we have:

j

iii

L

i

jisss

j xeaxp1

1)()|(

16

Case 1 (cont)

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

By the independence of the xj’s, p(x1,...,xn| θ)=∏jp(xj|θ).

Thus, if Akl = #(transitions from k to l) in the training set, and Ek(b) = #(emissions of symbol b from state k) in the training set, we have:

( )

( , ) ( , )

1 ( | ) [ ( )],.., kl kA E bkkl

k l k b

np a e bx x

17

Case 1 (cont)

( )

( , ) ( , )

[ ( )]kl kA E bkkl

k l k b

a e b

So we need to find akl’s and ek(b)’s which maximize:

Subject to:

])(,

)(,

0 [

1 and 1 states allFor

bea

beak

kkl

l bkkl

18

Case 1 (cont)

( )

( , ) ( , )

( )

[ ( )]

[ ( )][ ] [ ]

kl k

kl k

A E bkkl

k l k b

A E bkkl

k l k b

F a e b

a e b

kb

Subject to: for all , 1, and e ( ) 1.kll

k a b

Rewriting, we need to maximize:

19

Case 1 (cont)

If we maximize for each : s.t. 1klAklkl

ll

k a a ( )and also [ ( )] s.t. ( ) 1kE b

k kbb

e b e b

Then we will maximize also F.Each of the above is a simpler ML problem, which is similar to ML parameters estimation for a die, treated next.

20

ML parameters estimation for a die

Let X be a random variable with 6 values x1,…,x6

denoting the six outcomes of a die. Here the parameters areθ ={1,2,3,4,5, 6} , ∑θi=1Assume that the data is one sequence:

Data = (x6,x1,x1,x3,x2,x2,x3,x4,x5,x2,x6)So we have to maximize

2 3 2 21 2 3 4 5 6( | )P Data

Subject to: θ1+θ2+ θ3+ θ4+ θ5+ θ6=1 [and θi 0 ]

252 3 2

1 2 3 4 51

i.e., ( | ) 1 ii

P Data

23

Maximum Likelihood EstimateBy the ML approach, we look for parameters that maximizes the probability of data (i.e., the likelihood function ).We will find the parameters by considering the corresponding log-likelihood function:

63 51 2 4

551 2 3 4 1

log ( | ) log 1N

N NN N Nii

P Data

5

16

5

11loglog

i ii ii NN

A necessary condition for (local) maximum is:

01

)|(log5

1

6

θ

i ij

j

j

NNDataP

24

Finding the Maximum

Rearranging terms:

ii

jj N

N Divide the jth equation by the ith

equation:

Sum from j=1 to 6:

ii

ii

j j

jj N

N

N

N

6

16

1

1

So there is only one local – and hence global – maximum. Hence the MLE is given by:

6,..,1 iN

N ii

6

65

1

6

1 NNN

i ij

j

25

Generalization for distribution with any number n of outcomes

Let X be a random variable with n values x1,…,xk denoting the k outcomes of Independently and Identically Distributed experiments, with parameters θ ={1,2,...,k} (θi is the probability of xi). Again, the data is one sequence of length n:

Data = (xi1,xi2

,...,xin)

Then we have to maximize

1 211 2( | ) , ( ... )knn n

kkP Data n n n

Subject to: θ1+θ2+ ....+ θk=1

11

1

1 11

i.e., ( | ) 1k

k

nknn

iki

P Data

26

Generalization for n outcomes (cont)

i k

i k

n n

By treatment identical to the die case, the maximum is obtained when for all i:

Hence the MLE is given by the relative frequencies:

1,..,ii

ni k

n

27

Fractional Exponents

Some models allow ni’s to be fractions (eg, if we are uncertain of a die outcome, we may consider it “6” with 20% confidence and “5” with 80%):We still can have

And the same analysis yields:

1,..,ii

ni k

n

1 211 2( | ) , ( ... )knn n

kkP Data n n n

28

Apply the ML method to HMM

Let Akl = #(transitions from k to l) in the training set.Ek(b) = #(emissions of symbol b from state k) in the training set. We need to:

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

( )

( , ) ( , )

Maximize [ ( )]kl kA E bkkl

k l k b

a e b

k Subject to: for all states , 1, and e ( ) 1, , ( ) 0.kl kl kl b

k a b a e b

29

Apply to HMM (cont.)

We apply the previous technique to get for each k the parameters {akl|l=1,..,m} and {ek(b)|bΣ}:

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

'' '

( ) , and ( )

( ')kl k

kl kkl kl b

A E ba e b

A E b

Which gives the optimal ML parameters

30

Summary of Case 1: Sequences are fully known

We know the complete structure of each sequence in the training set {x1,...,xn}. We wish to estimate akl and ek(b) for all pairs of states k, l and symbols b.

Summary: when everything is known, we can find the (unique set of) parameters θ* which maximizes

p(x1,...,xn| θ*) =MAXθ p(x1,...,xn| θ).

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

31

Adding pseudo counts in HMM

If the sample set is too small, we may get a biased result. In this case we modify the actual count by our prior knowledge/belief: rkl is our prior belief and transitions from k to l.rk(b) is our prior belief on emissions of b from state k.

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

' '' '

( ) ( )then , and ( )

( ) ( ( ') ( ))kl kl k k

kl kkl kl k kl b

A r E b r ba e b

A r E b r b

32

Case 2: State paths are unknown

For a given θ we have: p(x1,..., xn|θ)= p(x1| θ) p (xn|θ)

(since the xj are independent)

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

For each sequence x,p(x|θ)=∑s p(x,s|θ),

The sum taken over all state paths s which emit x.

33

Case 2: State paths are unknown

Thus, for the n sequences (x1,..., xn) we have:

p(x1,..., xn |θ)= ∑ p(x1,..., xn , s1,..., sn |θ),

Where the summation is taken over all tuples of n state paths (s1,..., sn ) which generate (x1,..., xn ) .For simplicity, we will assume that n=1.

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

(s1,..., sn )

34

Case 2: State paths are unknown

So we need to maximize p(x|θ)=∑s p(x,s|θ), where the summation is over all the sequences S which produce the output sequence x. Finding θ* which maximizes ∑s p(x,s|θ) is hard. [Unlike finding θ* which maximizes p(x,s|θ) for a single sequence (x,s).]

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

35

ML Parameter Estimation for HMM

The general process for finding θ in this case is1. Start with an initial value of θ.2. Find θ’ so that p(x|θ’) > p(x|θ) 3. set θ = θ’.4. Repeat until some convergence criterion is met.

A general algorithm of this type is the Expectation Maximization algorithm, which we will meet later. For the specific case of HMM, it is the Baum-Welch training.

36

Baum Welch training

We start with some values of akl and ek(b), which define prior values of θ. Then we use an iterative algorithm which attempts to replace θ by a θ* s.t.

p(x|θ*) > p(x|θ)This is done by “imitating” the algorithm for Case 1, where all states are known:

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

37

Baum Welch training

In case 1 we computed the optimal values of akl and ek(b), (for the optimal θ) by simply counting the number Akl of transitions from state k to state l, and the number Ek(b) of emissions of symbol b from state k, in the training set. This was possible since we knew all the states.

Si= lSi-1= k

xi-1= b

… …

xi= c

38

Baum Welch training

When the states are unknown, the counting process is replaced by averaging process:For each edge si-1 si we compute the average number of “k to l” transitions, for all possible pairs (k,l), over this edge. Then, for each k and l, we take Akl to be the sum over all edges.

Si= ?Si-1= ?

xi-1= b xi= c

… …

39

Baum Welch training

Similarly, For each edge si b and each state k, we compute the average number of times that si=k, which is the expected number of “k → b” transmission on this edge. Then we take Ek(b) to be the sum over all such edges. These expected values are computed by assuming the current parameters θ:

Si = ?

xi-1= b

40

Akl and Ek(b) when states are unknown

Akl and Ek(b) are computed according to the current distribution θ, that is:

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi

Akl=∑sAsklp(s|x,θ), where As

kl is the number of k to l transitions in the sequence s.Ek(b)=∑sEs

k (b)p(s|x,θ), where Esk (b) is the number of times k

emits b in the sequence s with output x.

41

Baum Welch: step 1a Count expected number of state transitions

For each i, k,l, compute the state transitions probabilities by the current θ:

s1 SisL

X1 Xi XL

Si-1

Xi-1.. ..

P(si-1=k, si=l | x,θ)For this, we use the forwards and backwards algorithms

43

Baum Welch: Step 1a (cont)

Claim: By the probability distribution of HMM

s1 SisL

X1 Xi XL

Si-1

Xi-1.. ..

1

1( ) ( ) ( )( , | , )

( | )k kl l i l

i i

F i a e x B ip s k s l x

p x

(akl and el(xi) are the parameters defined by , and Fk(i-1), Bk(i) are the forward and backward algorithms)

44

Step 1a: Computing P(si-1=k, si=l | x,θ)

P(x1,…,xL,si-1=k,si=l|) = P(x1,…,xi-1,si-1=k|) aklel(xi ) P(xi+1,…,xL |si=l,)

= Fk(i-1) aklel(xi ) Bl(i)

Via the forward algorithm

Via the backward algorithm

s1 s2 sL-1 sL

X1 X2 XL-1 XL

Si-1

Xi-1

si

Xi

x

p(si-1=k,si=l | x, ) = Fk(i-1) aklel(xi ) Bl(i)

)|( xp

45

Step 1a (end)

For each pair (k,l), compute the expected number of state transitions from k to l, as the sum of the expected number of k to l transitions over all L edges :

1

1

1

1

11

( , , | )( | )

( ) ( ) ( )( | )

L

kl i i

i

L

k kl l i l

i

A p s k s l xp x

F i a e x B ip x

46

Step 1a for many sequences:

Exercise: Prove that when we have n independent input sequences (x1,..., xn ), then Akl is given by:

11 1

1 1

1( = , = , )

( )

1( 1) ( ) ( )

( )

|n L

kl i ijj i

n Lj j

kl l ik ljj i

jA p s k s lp x

f i a e x b ip x

x

where and are the forward and backward algorithms

for under .

j jk lf b

jx

47

Baum-Welch: Step 1b count expected number of symbols emissions

for state k and each symbol b, for each i where Xi=b, compute the expected number of times that Xi=b.

s1 s2 sL-1 sL

X1 X2 XL-1 XL

si

Xi=b

),...,(/)()(

),...(/),...(

),...|(

Likik

LiL

Li

xxpsbsf

xxpksxxp

xxksp

1

11

1

48

Baum-Welch: Step 1b

For each state k and each symbol b, compute the expected number of emissions of b from k as the sum of the expected number of times that si = k, over all i’s for which xi = b.

1

:

( ) ( ) ( )( | )

i

k k k

i x b

E b F i B ip x

49

Step 1b for many sequences

Exercise: when we have n sequences (x1,..., xn ), the expected number of emissions of b from k is given by:

1 :

1( ) ( ) ( )

( ) ji

nj j

k k kjj i x b

E b F i B ip x

50

Summary of Steps 1a and 1b: the E part of the Baum Welch training

These steps compute the expected numbers Akl of k,l transitions for all pairs of states k and l, and the expected numbers Ek(b) of transmitions of symbol b from state k, for all states k and symbols b.

The next step is the M step, which is identical to the computation of optimal ML parameters when all states are known.

51

Baum-Welch: step 2

'' '

( ) , and ( )

( ')kl k

kl kkl kl b

A E ba e b

A E b

Use the Akl’s, Ek(b)’s to compute the new values of akl and ek(b). These values define θ*.

The correctness of the EM algorithm implies that: p(x1,..., xn|θ*) p(x1,..., xn|θ)

i.e, θ* increases the probability of the data

This procedure is iterated, until some convergence criterion is met.