Transcript GS 540 Discussion section 7 HW6: Baum-Welch • Goal: learn HMM parameters taking into account all paths: • Expectation maximization – Forward backward algorithm. – Re-estimate.

GS 540
Discussion section 7
HW6: Baum-Welch
• Goal: learn HMM parameters taking into
account all paths:
• Expectation maximization
– Forward backward algorithm.
– Re-estimate parameter values based on expected
counts.
Likelihood function for Viterbi and
EM
• Likelihood function for Viterbi:
• Likelihood function for EM:
Baum-Welch
1. Use forward algorithm to find log likelihood
of the sequence (ie. sum of all paths)
2. Use forward-backward to get fractional
counts for each edge type
– (total prob of paths passing through edge)/(total
prob of all paths)
3. Re-estimate transition and emission probs
by calculating the expected number of each
edge type
Forward-backward algorithm
sequence
states
A
C
T
A
C
T
T
…
Store at each node:
• Forward: Sum of probabilities of paths ending at
position i state k.
• Backward: Sum of probabilities of path starting at
position i state k.
Forward-backward algorithm
sequence
states
A
C
T
A
C
T
T
…
forward(i,k) =
sum_k' [ forward(i-1,k') * transition(k,k') *
emission(S_i-1, k') ]
Forward-backward algorithm
sequence
states
A
C
T
A
C
T
T
…
backward(i,k) =
sum_k' [ backward(i+1,k') * transition(k,k') *
emission(S_i+1, k') ]
Forward-backward algorithm
sequence
states
A
C
T
A
C
T
T
…
Total probability of paths passing through position i
state k:
forward(i, k) * backward(i, k) * emission(S_i, k)
Use this to update emission(X_i, k)
Forward-backward algorithm
sequence
states
A
C
T
A
C
T
T
…
Total probability of paths passing through position i-1 state
k' to position i state k:
forward(i-1, k') * backward(i, k) *
emission(S_i-1, k') * emission(S_i, k) * transition(k',k)
Use this to update transition(k', k)
Numeric stability in Baum-Welch
• Baum-Welch involves addition of probabilities,
so using log space is not trivial.
• Two strategies for numeric stability:
– Multiply probabilities by a constant factor at each
position (not recommended).
– Use logs, and implement log(A+B) carefully
(recommended).
Addition in log space
• Not stable:
• More stable:
Addition in log space
• More stable:
Be careful with log(0)
• Use a special value for log(0)
• log(LOGZERO + p) = log(p)
• log(LOGZERO * p) = LOGZERO
HW6 Tips
• Carry out algorithm by hand for a few steps.
• Compute likelihood at each Baum-Welch
iteration. If it goes down, you have a bug.
HW7
• HMM to predict genes
– 11 states
– Each emits trinucleotides (codons)
• Viterbi parse
• 5 iterations of training
UCSC Genome Browser