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# phylo1 - HMM Training Baum-Welch Algorithm Christopher Lee...

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HMM Training: Baum-Welch Algorithm Christopher Lee December 1, 2009

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How to model gene evolution? atggggctcagcgacggggagtggcagcaggtgctgaacgtctgggggaa atggggctcagtgatggggagtggcagatggtgctgaacatctgggggaa atggctgatcatgatctggttctgaagtgctggggagccgtggaggccga atggctaactatgacatggttctgcagtgctgggggccagtggaggctga 1
Evolution as a Markov Chain? Presumably, evolution from a given ancestor A depends only on sequence of A , not on its ancestors. But discrete time assumption no longer tenable. 2

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Continuous Time Markov Chains For a homogeneous process; define state probability vec- tor π ( t ) and transition matrix at time t as T ( t ) : π ( t ) = π ( t = 0 ) T ( t ) T ( t + Δ t ) = T ( t ) T ( Δ t ) Define instantaneous rate matrix Λ : Λ = lim Δ t 0 T ( Δ t ) - I Δ t 3
Matrix Exponential We can then calculate T ( t ) as T ( t ) = e Λ t where for a square matrix M e M = I + M + M 2 2! + ... + M i i ! + ... = i = 0 M i i ! Elegant, but not easy to calculate generally... 4

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Simple Mutation Model Assumptions Neutral: mutation only; no selection Reversible: π i λ i j = π j λ ji Independence: λ i j = π j μ 5
F81 Model (Felsenstein)

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