CSE182-L10

1
CSE182-L10

• HMM applications

2
Probability of being in specific states

• What is the probability that we were in state k
  at step I?
• PrAll paths that passed through state k
  at step I, and emitted x
• PrAll paths that
  emitted x

3
The Forward Algorithm

• Recall vi,j Probability of the most likely
  path the automaton chose in emitting x1xi, and
  ending up in state j.
• Define fi,j Probability that the automaton
  started from state 1, and emitted x1xi
• What is the difference?

x1xi
4
Most Likely path versus Probability of Arrival

• There are multiple paths from states 1..j in
  which the automaton can output x1xi
• In computing the viterbi path, we choose the most
  likely path
• Vi,j maxp Prx1xip
• The probability of emitting x1xi and ending up
  in state j is given by
• Fi,j ?p Prx1xip

5
The Forward Algorithm

• Recall that
• v(i,j) max l?Q v(i-1,l).Al,j .ej(xi)
• Instead
• F(i,j) ?l?Q (F(i-1,l).Al,j ).ej(xi)

1
j
6
The Backward Algorithm

• Define bi,j Probability that the automaton
  started from state i, emitted xi1xn and ended
  up in the final state

xi1xn
x1xi
1
m
i
7
Forward Backward Scoring

• F(i,j) ?l?Q (F(i-1,l).Al,j ).ej(xi)
• Bi,j ?l?Q (Aj,l.el(xi1) B(i1,l))
• Prx,pikF(i,k) B(i,k)

8
Application of HMMs

• How do we modify this to handle indels?

9
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
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
A C G T
10
Profile HMMs

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

1 2 3 4 5 6 7 8
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
A C G T
C
A
A
A
T
G
T
C
11
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) and vDj(i) are defined similarly.

12
Viterbi Equations for Profile HMMs
vMj-1(i-1) log(AMj-1, Mj) vMj(i)
log (eMj(xi)) max vIj-1(i-1)
log(AIj-1, Mj)
vDj-1(i-1) log(ADj-1,
Mj)
vMj(i-1) log(AMj-1, Ij) vIj(i)
log (eIj(xi)) max vIj(i-1)
log(AIj-1, Ij)
vDj(i-1) log(ADj-1, Ij)
13
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?

14
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
0.1
.25
end
start
C
T
0.4
.25
15
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
T
0.25
16
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.

17
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.