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Estimating Phylogenies with Maximum Likelihood Methods

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This document explains the application of maximum likelihood methods in estimating phylogenies based on specific evolutionary models. It details the likelihood calculations using coin data and tree structures, incorporating the probabilities of observed data at various genetic sites. By analyzing the likelihood of evolutionary trees, we derive maximum probabilities for given sequences, facilitating a deeper understanding of evolutionary relationships. The approach demonstrates the independent evolution of different sites and lineages, emphasizing the mathematical foundations behind phylogenetic tree estimation.

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Estimating Phylogenies with Maximum 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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