Thomas Jellema & Wouter Van Gool 1 Question. 2Answer.

Thomas Jellema & Wouter Van Gool 1

QuestionQuestion


AnswerAnswer


Pairwise alignment using Pairwise alignment using HMMsHMMs

Wouter van Gool and Thomas Jellema


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



4.1 Most probable path4.1 Most probable path

Model that emits a single sequene



Begin and end state



Model that emits a pairwise alignment



Example of a sequenceSeq1: A C T _ CSeq2: T _ G G CAll : M X M Y M



Begin and end state



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



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



Probability of going to the M state



Viterbi algorithm for pair HMMs



Finding the most probable path using FSAs

-The most probable path is also the optimal FSA alignment



Finding the most probable path using FSAs



Recurrence relations



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



Random model


1END

η1- ηY

η1- ηX

ENDYX

1END

τε1-ε -τ

Y

τ ε1-ε -τX

τδδ1-2δ -τ

M

ENDYXM

“Model”

“Random”




Transitions


4.1 Most probable path4.1 Most probable pathOptimal log-odds alignment


4.1 Most probable path4.1 Most probable pathA pair HMM for local alignment


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



4.2 Probability of an allignment4.2 Probability of an allignment

Probability that a given pair of sequences are related.



Summing the probabilities


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



4.3 Suboptimal alignment4.3 Suboptimal alignment

Finding suboptimal alignments

How to make sample alignments?


4.3 Suboptimal alignment4.3 Suboptimal alignmentFinding distinct suboptimal alignments


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



Contents




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))


Posterior probability that xPosterior probability that x ii is is

aligned to yaligned to yii

Notation: xi ◊ yi means xi is aligned to yi


Posterior probability that xPosterior probability that x ii is is

aligned to yaligned to yii


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



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


Contents




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


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


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



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



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



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.


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



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


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.


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!


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


Thomas Jellema & Wouter Van Gool 1 Question. 2Answer.

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