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### CSE182-L10

HMM applications

Probability of being in specific states

- What is the probability that we were in state k at step I?

Pr[All paths that passed through state k at step I, and emitted x]

Pr[All paths that emitted x]

The Forward Algorithm

- Recall v[i,j] : Probability of the most likely path the automaton chose in emitting x1…xi, and ending up in state j.
- Define f[i,j]: Probability that the automaton started from state 1, and emitted x1…xi
- What is the difference?

x1…xi

Most Likely path versus Probability of Arrival

- There are multiple paths from states 1..j in which the automaton can output x1…xi
- In computing the viterbi path, we choose the most likely path
- V[i,j] = maxπ Pr[x1…xi|π]
- The probability of emitting x1…xi and ending up in state j is given by
- F[i,j] = ∑π Pr[x1…xi|π]

The Forward Algorithm

- Recall that
- v(i,j) = max lQ {v(i-1,l).A[l,j] }.ej(xi)
- Instead
- F(i,j) = ∑lQ (F(i-1,l).A[l,j] ).ej(xi)

1

j

The Backward Algorithm

- Define b[i,j]: Probability that the automaton started from state i, emitted xi+1…xn and ended up in the final state

xi+1…xn

x1…xi

1

m

i

Forward Backward Scoring

- F(i,j) = ∑lQ (F(i-1,l).A[l,j] ).ej(xi)
- B[i,j] = ∑lQ (A[j,l].el(xi+1) B(i+1,l))
- Pr[x,πi=k]=F(i,k) B(i,k)

A

C

G

T

0.9 0.4 0.3 0.6 0.1 0.0 0.2 1.0

0.0 0.2 0.7 0.0 0.3 0.0 0.0 0.0

0.1 0.2 0.0 0.0 0.3 1.0 0.3 0.0

0.0 0.2 0.0 0.4 0.3 0.0 0.5 0.0

Application of HMMs- How do we modify this to handle indels?

Applications of the HMM paradigm

- Modifying Profile HMMs to handle indels
- States Ii: insertion states
- States Di: deletion states

1 2 3 4 5 6 7 8

A

C

G

T

0.9 0.4 0.3 0.6 0.1 0.0 0.2 1.0

0.0 0.2 0.7 0.0 0.3 0.0 0.0 0.0

0.1 0.2 0.0 0.0 0.3 1.0 0.3 0.0

0.0 0.2 0.0 0.4 0.3 0.0 0.5 0.0

Profile HMMs

- An assignment of states implies insertion, match, or deletion. EX: ACACTGTA

1 2 3 4 5 6 7 8

A

C

G

T

0.9 0.4 0.3 0.6 0.1 0.0 0.2 1.0

0.0 0.2 0.7 0.0 0.3 0.0 0.0 0.0

0.1 0.2 0.0 0.0 0.3 1.0 0.3 0.0

0.0 0.2 0.0 0.4 0.3 0.0 0.5 0.0

C

A

A

C

T

G

T

A

Viterbi Algorithm revisited

- Define vMj (i)as the log likelihood score of the best path for matching x1..xi to profile HMM ending with xi emitted by the state Mj.
- vIj(i)andvDj(i)are defined similarly.

Viterbi Equations for Profile HMMs

vMj-1(i-1) + log(A[Mj-1, Mj])

vMj(i) = log (eMj(xi)) + max vIj-1(i-1) + log(A[Ij-1, Mj])

vDj-1(i-1) + log(A[Dj-1, Mj])

vMj(i-1) + log(A[Mj-1, Ij])

vIj(i) = log (eIj(xi)) + max vIj(i-1) + log(A[Ij-1, Ij])

vDj(i-1) + log(A[Dj-1, Ij])

Compositional Signals

- CpG islands. In genomic sequence, the CG di-nucleotide is rarely seen
- CG helps methylation of C, and subsequent mutation to T.
- In regions around a gene, the methylation is suppressed, and therefore CG is more common.
- CpG islands: Islands of CG on the genome.
- How can you detect CpG islands?

An HMM for Genomic regions

- Node A emits A with Prob. 1, and 0 for all other bases.
- The start and end node do not emit any symbol.
- All outgoing edges from nodes are equi-probable, except for the ones coming out of C.

A

G

.25

0.1

end

start

C

0.4

T

.25

An HMM for CpG islands

- Node A emits A with Prob. 1, and 0 for all other bases.
- The start and end node do not emit any symbol.
- All outgoing edges from nodes are equi-probable, except for the ones coming out of C.

A

G

0.25

0.25

end

start

C

0.25

T

HMM for detecting CpG Islands

A

B

A

G

A

0.1

end

G

start

end

C

start

0.4

T

C

T

- In the best parse of a genomic sequence, each base is assigned a state from the sets A, and B.
- Any substring with multiple states coming from B can be described as a CpG island.

HMM: Summary

- HMMs are a natural technique for modeling many biological domains.
- They can capture position dependent, and also compositional properties.
- HMMs have been very useful in an important Bioinformatics application: gene finding.

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