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Likelihood methods

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Likelihood methods. Given a particular model of evolution , we can estimate phylogenies using maximum likelihood. Likelihood with a coin. data ( D ). HHTTHTHHTHHTTT. L = Prob ( HHTTHTHHTHHTTT | p ) = p p (1-p ) (1-p ) p (1-p ) p p (1-p ) p p (1-p ) (1-p ) (1-p). = p 5 (1-p) 6.

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Likelihood methods

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  1. Likelihoodmethods Given a particular model of evolution, we can estimate phylogenies using maximum likelihood

  2. Likelihood with a coin data (D) HHTTHTHHTHHTTT L = Prob (HHTTHTHHTHHTTT | p) = p p (1-p) (1-p) p (1-p) p p (1-p) p p (1-p) (1-p) (1-p) = p5(1-p)6 likelihood

  3. Likelihood with a coin likelihoodcurve p=0.4545…

  4. Likelihood with a coin L = Prob (HHTTHTHHTHHTTT | p) = p5(1-p)6 dL dp = 5p4(1-p)6 – 6p5(1-p)5  L maximal if 5p4(1-p)6 – 6p5(1-p)5 = 0  p = 5/11

  5. Likelihood with a coin L = Prob (HHTTHTHHTHHTTT | p) = p5(1-p)6 ln(L) = 5 ln(p) + 6 ln(1-p)  ln(L) maximal if  p = 5/11 5 6 - = 0 p (1-p)

  6. Likelihood of a tree Given a tree topology and branch lengths an evolutionary model (sequence) data A G C T and assuming evolution in different sites is independent evolution in different lineages is independent

  7. Likelihood of a tree m L = Prob(D|T) = ∏ Prob(Di|T) i=1 data at the ithsite

  8. Likelihood of a tree G C t4 t5 C A A C t2 t7 t3 t1 A A t6 t8 A

  9. Likelihood of a tree Prob(Di|T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T) x y z w ancestor x can be A, C, T or G likelihood of the observed data at site i

  10. Likelihood of a tree Prob(A, C, C, C, G, x, y, z, w | T) = Prob(x) … C G t4 t5 C w A C t2 t3 t7 t1 z y t8 t6 x

  11. Likelihood of a tree Prob(A, C, C, C, G, x, y, z, w | T) = Prob(x) • Prob (y|x, t6) … C G t4 t5 C w A C t2 t3 t7 t1 z y t8 t6 x

  12. Likelihood of a tree Prob(A, C, C, C, G, x, y, z, w | T) = Prob(x) • Prob (y|x, t6) • Prob (A|y, t1) … C G t4 t5 C w A C t2 t3 t7 t1 z y t8 t6 x

  13. Likelihood of a tree Prob(A, C, C, C, G, x, y, z, w | T) = Prob(x) • Prob (y|x, t6) • Prob (A|y, t1) • Prob (C|y, t2) • Prob (z|x, t8) • Prob (C|z, t3) • Prob (w|z, t7) • Prob (C|w, t4) • Prob (G|w, t5) C G t4 t5 C w A C t2 t3 t7 t1 z y t8 t6 x

  14. Likelihood of a tree Prob(Di|T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T) x y z w

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