1. Learning HMM parameters
Sushmita Roy
BMI/CS 576
Oct 31st, 2013

2. Recall the three questions in HMMs
Given a sequence of observations how likely is it an HMM to have generated it?
Forward algorithm
What is the most likely sequence of states that has generated a sequence of observations
Viterbi
How can we learn an HMM from a set of sequences?
Forward-backward or Baum-Welch (an EM algorithm)

3. Learning HMMs from data
Parameter estimation
If we knew the state sequence it would be easy to estimate the parameters
But we need to work with hidden state sequences
Use “expected” counts of state transitions

4. Learning without hidden information
Learning is simple if we know the correct path for each sequence in our training set 2 2 4 4 5
C A G T
begin 1 3
end 5 2 4
Estimate parameters by counting the number of times each parameter is used across the training set

5. Learning without hidden information
Transition probabilities
Emission probabilities
Number of transitions from k to l
k,l are states
Number of times b is emitted from k

6. Learning with hidden information
if we don’t know the correct path for each sequence in our training set, consider all possible paths for the sequence ? ? ? ? 5
C A G T
begin 1 3
end 5 2 4
estimate parameters through a procedure that counts the expected number of times each parameter is used across the training set

7. The Baum-Welch algorithm
Also known as Forward-backward algorithm
An Expectation Maximization algorithm
Expectation: Estimate the “expected” number of times there are transitions and emissions (using current values of parameters)
Maximization: Estimate parameters given hidden variables
Hidden variables are the state transitions and emission counts

8. The expectation step
We need to know the probability of the i th symbol being produced by state k, given sequence x (posterior probability of state k at time t)
We also need to know the probability of ith and (i+1)th symbol being produced by state k, and l given sequence x
Given these we can compute our expected counts for state transitions, character emissions

9. Computing We will do this in a somewhat indirect manner
First we compute the probability of the entire observed sequence with the tth symbol being generated by state k
Forward algorithm fk(t)
Backward algorithm bk(t)

10. Computing
If we can compute
How can we get
Forward step

11. The backward algorithm
the backward algorithm gives us , the probability of observing the rest of x, given that we’re in state k after i characters
0.4
0.2
A 0.4
C 0.1
G 0.2
T 0.3
A 0.2
C 0.3
G 0.3
T 0.2
0.8
0.6
0.5
begin 1 3
end 5
0.5
A 0.4
C 0.1
G 0.1
T 0.4
A 0.1
C 0.4
G 0.4
T 0.1
0.9
0.2 2 4
0.1
0.8
C A G T

12. Steps of the backward algorithm
Initialization (t=T)
Recursion (t=T-1 to 1)
Termination

13. Computing
This is

14. Putting it all together
We need the expected number of times c is emitted by state k
And the expected number of times k transitions to l
Training sequences

15. The maximization step Estimate new emission parameters by:
Estimate new transition parameters by
Just like in the simple case but typically we’ll do some “smoothing” (e.g. add pseudocounts)

16. The Baum-Welch algorithm
initialize the parameters of the HMM
iterate until convergence
initialize , with pseudocounts
E-step: for each training set sequence j = 1…n
calculate values for sequence j
add the contribution of sequence j to ,
M-step: update the HMM parameters using ,

17. Baum-Welch algorithm example
given
the HMM with the parameters initialized as shown
the training sequences TAG, ACG
A 0.1
C 0.4
G 0.4
T 0.1
A 0.4
C 0.1
G 0.1
T 0.4
begin
end
1.0
0.1
0.9
0.2
0.8 3 2 1
we’ll work through one iteration of Baum-Welch

18. Baum-Welch example (cont)
Determining the forward values for TAG
Here we compute just the values that are needed for computing successive values.
For example, no point in calculating f1(3)
In a similar way, we also compute forward values for ACG

19. Baum-Welch example (cont)
Determining the backward values for TAG
Again, here we compute just the values that are needed
In a similar way, we also compute backward values for ACG

20. Baum-Welch example (cont)
determining the expected emission counts for state 1
contribution
of TAG
contribution
of ACG
pseudocount
*note that the forward/backward values in these two columns differ; in each column
they are computed for the sequence associated with the column

21. Baum-Welch example (cont)
determining the expected transition counts for state 1 (not using pseudocounts)
in a similar way, we also determine the expected emission/transition counts for state 2
Contribution of TAG
Contribution of ACG

22. Baum-Welch example (cont)
determining probabilities for state 1

23. Summary Three problems in HMMs Probability of an observed sequence
Forward algorithm
Most likely path for an observed sequence
Viterbi
Can be used for segmentation of observed sequence
Parameter estimation
Baum-Welch
The backward algorithm is used to compute a quantity needed to estimate the posterior of a state given the entire observed sequence