Likelihood methods
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Likelihood methods Given a particular model of evolution, we can estimate phylogenies using maximum likelihood
description
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. - PowerPoint PPT Presentation
Transcript of Likelihood methods
Likelihood methodsGiven 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 = p5(1-p)6
Likelihood with a coin
p
0.0 0.2 0.4 0.6 0.8 1.0
Like
lihoo
d
likelihood curve
p=0.4545…
Likelihood with a coin
L = Prob (HHTTHTHHTHHTTT | p) = p5(1-p)6
dLdp = 5p4(1-p)6 – 6p5(1-p)5
L maximal if 5p4(1-p)6 – 6p5(1-p)5 = 0
p = 5/11
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 6p (1-p)
- = 0
Likelihood of a tree
a tree topology and branch lengths
an evolutionary modelA G
C T
(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(Di|T)i=1
m
data at the ith site
Likelihood of a tree
A
A
A
AA
C
C
C
G
t1
t2
t6 t8
t3 t7
t4 t5
Likelihood of a tree
Prob(Di|T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T)x
ancestor x can be A, C, T or G
y z w
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
wA C
C
C
G
t1
t2
t6 t8
t3 t7
t4 t5
Likelihood of a tree
Prob (A, C, C, C, G, x, y, z, w | T) =
Prob(x) • Prob (y|x, t6) …
y
x
z
wA C
C
C
G
t1
t2
t6 t8
t3 t7
t4 t5
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) …
y
x
z
wA C
C
C
G
t1
t2
t6 t8
t3 t7
t4 t5
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)
y
x
z
wA C
C
C
G
t1
t2
t6 t8
t3 t7
t4 t5
Likelihood of a tree
#tip species # interior nodes # scenarios
4 3 645 4 256
10 9 26214420 19 27487790694425 24 281474976710656
Prob(Di|T) = ∑ ∑ ∑ ∑ Prob (A, C, C, C, G, x, y, z, w | T)x y z w
