Likelihood methods Given a particular model of evolution, we can estimate phylogenies using maximum likelihood.

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Presentation transcript:

Likelihood methods Given a particular model of evolution, we can estimate phylogenies using maximum likelihood

Likelihood with a coin 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) HHTTHTHHTHHTTT data (D) likelihood = p 5 (1-p) 6

Likelihood with a coin likelihood curve p=0.4545…

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

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

Likelihood of a tree a tree topology and branch lengths an evolutionary model (sequence) data Given and assuming 1.evolution in different sites is independent 2.evolution in different lineages is independent

Likelihood of a tree L = Prob(D|T) = ∏ Prob(D i |T) i=1 m data at the i th site

Likelihood of a tree A A A A A C C C G t1t1 t2t2 t6t6 t8t8 t3t3 t7t7 t4t4 t5t5

Prob(D i |T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T) x ancestor x can be A, C, T or G yzw likelihood of the observed data at site i

Likelihood of a tree Prob (A, C, C, C, G, x, y, z, w | T) = Prob(x) … y x z w A C C C G t1t1 t2t2 t6t6 t8t8 t3t3 t7t7 t4t4 t5t5

Likelihood of a tree Prob (A, C, C, C, G, x, y, z, w | T) = Prob(x) Prob (y|x, t 6 ) … y x z w A C C C G t1t1 t2t2 t6t6 t8t8 t3t3 t7t7 t4t4 t5t5

Likelihood of a tree Prob (A, C, C, C, G, x, y, z, w | T) = Prob(x) Prob (y|x, t 6 ) Prob (A|y, t 1 ) … y x z w A C C C G t1t1 t2t2 t6t6 t8t8 t3t3 t7t7 t4t4 t5t5

Likelihood of a tree Prob (A, C, C, C, G, x, y, z, w | T) = Prob(x) Prob (y|x, t 6 ) Prob (A|y, t 1 ) Prob (C|y, t 2 ) Prob (z|x, t 8 ) Prob (C|z, t 3 ) Prob (w|z, t 7 ) Prob (C|w, t 4 ) Prob (G|w, t 5 ) y x z w A C C C G t1t1 t2t2 t6t6 t8t8 t3t3 t7t7 t4t4 t5t5

Likelihood of a tree #tip species# interior nodes# scenarios Prob(D i |T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T) xyzw