Thomas Jellema & Wouter Van Gool 1 Question. 2Answer.
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Thomas Jellema & Wouter Van Gool 1
QuestionQuestion
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AnswerAnswer
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Pairwise alignment using Pairwise alignment using HMMsHMMs
Wouter van Gool and Thomas Jellema
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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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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4.1 Most probable path4.1 Most probable path
Model that emits a single sequene
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4.1 Most probable path4.1 Most probable path
Begin and end state
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4.1 Most probable path4.1 Most probable path
Model that emits a pairwise alignment
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4.1 Most probable path4.1 Most probable path
Example of a sequenceSeq1: A C T _ CSeq2: T _ G G CAll : M X M Y M
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4.1 Most probable path4.1 Most probable path
Begin and end state
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4.1 Most probable path4.1 Most probable path
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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4.1 Most probable path4.1 Most probable path
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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4.1 Most probable path4.1 Most probable path
Probability of going to the M state
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4.1 Most probable path4.1 Most probable path
Viterbi algorithm for pair HMMs
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4.1 Most probable path4.1 Most probable path
Finding the most probable path using FSAs
-The most probable path is also the optimal FSA alignment
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4.1 Most probable path4.1 Most probable path
Finding the most probable path using FSAs
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4.1 Most probable path4.1 Most probable path
Recurrence relations
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4.1 Most probable path4.1 Most probable path
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)>0 iff the probability that a and b are related by our model is larger than the probability that they are picked at random.
The log odds scoring function
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4.1 Most probable path4.1 Most probable path
Random model
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1END
η1- ηY
η1- ηX
ENDYX
1END
τε1-ε -τ
Y
τ ε1-ε -τX
τδδ1-2δ -τ
M
ENDYXM
“Model”
“Random”
4.1 Most probable path4.1 Most probable path
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4.1 Most probable path4.1 Most probable path
Transitions
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4.1 Most probable path4.1 Most probable path
Transitions
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4.1 Most probable path4.1 Most probable pathOptimal log-odds alignment
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4.1 Most probable path4.1 Most probable pathA 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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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4.2 Probability of an allignment4.2 Probability of an allignment
Probability that a given pair of sequences are related.
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4.2 Probability of an allignment4.2 Probability of an allignment
Summing the probabilities
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4.2 Probability of an allignment4.2 Probability of an allignment
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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 yiPosterior probability that xi is aligned to yi Wouter• Pair HMMs versus FSAs for searching Wouter• Conclusion and summary Wouter• Questions
Pairwise alignment using Pairwise alignment using HMMsHMMs
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4.3 Suboptimal alignment4.3 Suboptimal alignment
Finding suboptimal alignments
How to make sample alignments?
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4.3 Suboptimal alignment4.3 Suboptimal alignmentFinding 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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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Posterior probability that xPosterior probability that x ii is is aligned to yaligned to yii
Local accuracy of an alignment?Reliability measure for each part of an
alignmentHMM as a local alignment measureIdea: P(all alignments trough (xi,yi))
P(all alignments of (x,y))
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Posterior probability that xPosterior probability that x ii is is
aligned to yaligned to yii
Notation: xi ◊ yi means xi is aligned to yi
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Posterior probability that xPosterior probability that x ii is is aligned to yaligned to yii
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Posterior probability that xPosterior probability that x ii is is
aligned to yaligned to yii
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Probability alignmentProbability 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 > i1 and j1 < j2
Restriction to pairs (i,j) for which
P(xi ◊ yi )>0.5
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Posterior probability that xPosterior probability that x ii is is aligned to yaligned to yii
The expected accuracy of an alignment
Expected overlap between π and paths sampled from the posterior distribution
Dynamic programming
)1,(
),1(
)()1,1(
max),(
jiA
jiA
yxPjiA
jiAji
),(
)()(ji
ji yxPA
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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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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Pair HMMs versus FSAs for Pair HMMs versus FSAs for searchingsearching
P(D | M) > 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 Pair HMMs versus FSAs for searchingsearching
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 Pair HMMs vs FSAs for searchingsearching
Example: a model whose parameters match the data need not be the best model
a b a c
qa
S
B
α
1-α
1 1 1
1PS(abac) = α4qaqbqaqc
PB(abac) = 1-α
Model comparison using the best match rather than the total probability
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Pair HMMs vs FSAs for Pair HMMs vs FSAs for searchingsearching
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 Pair HMMs vs FSAs for searchingsearching
Bayesian model comparision: both HMMs have same log-odds ratio as previous FSA
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Pair HMMs vs FSAs for Pair HMMs vs FSAs for searchingsearching
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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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Why try to use HMMs?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 New things HMMs we can do with pair HMMswith 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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ConclusionConclusion
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
Pairwise alignment using Pairwise alignment using HMMsHMMs
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