Dia 1

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Model that emits a single sequene
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Begin and end state
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Model that emits a pairwise alignment
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Example of a sequence
Seq1 A C T _ C Seq2 T _ G G C All
M X M Y M
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Begin and end state
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Finding the most probable path

• The path you choose is the path that has the
  highest
• probability of being the correct alignment.
• The state we choose to be part of the alignment
  has to be
• the state with the highest probability of being
  correct.
• We calculate the probability of the state being
  a M, X or Y
• and choose the one with the highest probability
• If the probability of ending the alignment is
  higher
• then the next state being a M, X or Y then we
  end
• the alignment

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The probability of emmiting an M is the highest
probability of 1 previous state X new state
M 2 previous state Y new state M 3 previous
state M new state M
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Probability of going to the M state
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Viterbi algorithm for pair HMMs
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Finding the most probable path using FSAs
-The most probable path is also the optimal FSA
alignment
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Finding the most probable path using FSAs
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Recurrence relations
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The log odds scoring function

• We wish to know if the alignment score is above
  or below the score of random alignment.
• The log-odds ratio s(a,b) log (pab / qaqb).
• log (pab / qaqb)gt0 iff the probability that a
  and b are related by our model is larger than the
  probability that they are picked at random.

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Random model
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Random
Model
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Transitions
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Transitions
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Optimal log-odds alignment
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A pair HMM for local alignment
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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Probability that a given pair of sequences are
related.
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Summing the probabilities
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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Finding suboptimal alignments
How to make sample alignments?
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Finding distinct suboptimal alignments
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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Example Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion or summary Wouter
• Questions

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Posterior probability that xi is aligned to yi

• Local accuracy of an alignment?
• Reliability measure for each part of an alignment
• HMM as a local alignment measure
• Idea P(all alignments trough (xi,yi))
• P(all alignments of (x,y))

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Posterior probability that xi is aligned to yi

• Notation xi ? yi means xi is aligned to yi

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Posterior probability that xi is aligned to yi
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Posterior probability that xi is aligned to yi
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Probability alignment

• Miyazawa it seems attractive to find alignment
  by maximising P(xi ? yi )
• May lead to inconsistencies
• e.g. pairs (i1,i1) (i2,j2)
• i2 gt i1 and j1 lt j2
• Restriction to pairs (i,j) for which
• P(xi ? yi )gt0.5

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• Posterior probability that xi is aligned to yi
• The expected accuracy of an alignment
• Expected overlap between p and paths sampled from
  the posterior distribution
• Dynamic programming

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Pair HMMs versus FSAs for searching

• P(D M) gt P(M D)
• HMM maximum data likelihood by giving the same
  parameters (i.e. transition and emission
  probabilities)
• Bayesian model comparison with random model R

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Pair HMMs versus FSAs for searching

• Problems
• 1. Most algorithms do not compute full
  probability P(x,y M) but only best match
• or Viterbi path
• 2. FSA parameters may not be readily
  translated into probabilities

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Pair HMMs vs FSAs for searching

• Example a model whose parameters match the data
  need not be the best model

a
S
PS(abac) a4qaqbqaqc
1
1-a
PB(abac) 1-a
B
Model comparison using the best match rather than
the total probability
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1
1
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Pair HMMs vs FSAs for searching

• Problem no fixed scaling procedure can make the
  scores of this model into the log probabilities
  of an HMM

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Pair HMMs vs FSAs for searching

• Bayesian model comparision both HMMs have same
  log-odds ratio as previous FSA

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Pair HMMs vs FSAs for searching

• Conversion FSA into probabilistic model
• Probabilistic models may underperform standard
  alignment methods if Viterbi is used for database
  searching.
• Buf if forward algorithm is used, it would be
  better than standard methods.

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Example Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion and summary Wouter
• Questions

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Why try to use HMMs?

• Many complicated alignment algorithms
• can be described as simple Finite State
• Machines.
• HMMs have many advantages
• - Parameters can be trained to fit the data
  no need
• for PAM/BLOSSUM matrices
• - HMMs can keep track of all alignments, not
  just
• the best one

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New things HMMs we can do with pair HMMs

• Compute probability over all alignments.
• Compute relative probability of Viterbi
• alignment (or any other alignment).
• Sample over all alignments in proportion to their
  probability.
• Find distinct sub-optimal alignments.
• Compute reliability of each part of the best
• alignment.
• Compute the maximally reliable alignment.

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Conclusion

• Pairs-HMM work better for sequence alignment and
  database search than penalty score based
  alignment algorithms.
• Unfortunately both approaches are O(mn) and hence
  too slow for large database searches!

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• Contents
• Most probable path Thomas
• Probability of an alignment Thomas
• Sub-optimal alignments Thomas
• Pause
• Posterior probability that xi is aligned to yi
  Wouter
• Pair HMMs versus FSAs for searching Wouter
• Conclusion or summary Wouter
• Questions