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791_cb_lecture5 - 7.91 7.36 BE.490 Lecture#5 Mar 9 2004...

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Unformatted text preview: 7.91 / 7.36 / BE.490 Lecture #5 Mar. 9, 2004 Markov Models & DNA Sequence Evolution Chris Burge Review of Markov & HMM Models for DNA • Hidden Markov Models - looking under the hood Ch. 4 of Mount • Markov Models for splice sites • The Viterbi Algorithm • Real World HMMs CpG Islands %C+G 60 50 40 30 Hidden CpG Island Hidden Markov Model P P ig = 0.001 P = 0.99999 ii = 0.999 gg Genome P gi = 0.00001 Island … A C T C G A G T A CpG Island: C 0.3 G 0.3 A 0.2 T 0.2 Genome: 0.2 0.2 0.3 0.3 Observable CpG Island HMM II P = 0.99999 P ig = 0.001 Island gg “Transition Genome P ii = 0.999 probabilities” P gi = 0.00001 … T A C C G A G T A CpG Island: C G A T 0.3 0.3 0.2 0.2 Genome: 0.2 0.2 0.3 0.3 “Emission Probabilities” CpG Island HMM III … Want to infer A C T C G A G T A Observe But HMM is written in the other direction (observable depends on hidden) Inferring the Hidden from the Observable (Bayes’ Rule) P ( H = h 1 , h 2 ,..., h n | O = o 1 , o 2 ,..., o n ) Conditional Prob: P(A|B) = P(A,B)/P(B) P ( H = h 1 ,..., h n , O = o 1 , ..., o n ) = P ( O = o 1 , ..., o n ) P ( H = h 1 ,..., h n ) P ( O = o 1 ,..., o n | H = h 1 ,..., h n ) = P ( O = o 1 , ..., o n ) P ( O = o 1 ,..., o n ) somewhat difficult to calculate But notice : P ( H = h 1 , ..., h n , O = o 1 ,..., o n ) > P ( H = h ′ 1 , ..., h ′ n , O = o 1 , ..., o n ) implies P ( H = h 1 , ..., h n | O = o 1 ,..., o n ) > P ( H = h ′ 1 ,..., h ′ n | O = o 1 , ..., o n ) so can treat P ( O = o 1 ,..., o n ) as a constant Finding the Optimal “Parse” (Viterbi Algorithm) H opt opt opt , h 2 which maximizes joint probability: P ( H = h 1 , ..., h n , O = o 1 ,..., o n ) Want to find sequence of hidden states = h 1 opt , h 3 , ... (optimal “parse” of sequence) Solution: ( h ) = probability of optimal parse of the Define R i subsequence 1..i ending in state h ( h ) Solve recursively , i.e. determine R ( 2 h ) in terms of R 1 , etc. A. Viterbi, an MIT BS/MEng student in E.E. - founder of Qualcomm “Trellis” Diagram for Viterbi Algorithm Position in Sequence → 1 … i i+1 i+2 i+3 i+4 … L T … A T C G C … A Run time for k-state HMM on sequence of length L?...
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791_cb_lecture5 - 7.91 7.36 BE.490 Lecture#5 Mar 9 2004...

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