Likelihood ClassAnnotations March10.pdf - EEOB 6330 –...

EEOB 6330 – Spring 2020
Week 7 - March 3, 2020
Topic: Likelihood
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History of maximum likelihood in phylogenetics
I
Felsenstein (1981) is often cited as the first maximum likelihood method for
inferring phylogenies
I
But ....
I
Edwards and Cavalli-Sforza (1964) and Cavalli-Sforza and Edwards (1967)
provide important contributions to viewing estimation of phylogeny as a
problem in statistical inference
I
Edwards (1970) - maximum likelihood inference using gene frequency data
I
Neyman (1971) - maximum likelihood for sequence data under a simple
substitution model for three taxa
I
Felsenstein (1973) - maximum likelihood for discrete character data
2 / 43

Model-based phylogenetic inference
I
Motivation: Consider the evolutionary process along a phylogeny, and develop a
probabilistic model for the data observed at the tips of the tree
I
Begin by considering DNA sequence data, but models can be used for trait data
as well
I
The branch lengths will now have meaning for us – they can be interpreted as
proportional to the amount of evolutionary change
I
Start by considering the mutation process on one branch – model the probability
that one branch changes to another over a time unit of length t – What are some
important properties of this probability?
3 / 43
-
PAC
(
o
)
=
O
Paa
lo
)
=
I
-
proportional
to
the
length
t
:
A
•
£ -00
C
as
t
T
,
PAA
le
)
t
want
:
Pack
)
=
probability
that
A change
,
to cover
a
brink
of
length
t

Markov models of substitution
I
The most common models are:
I
finite-state
I
continuous-time
I
Markov models
I
We also commonly assume homogeneity across branches
I
We define these models by specifying an instantaneous rate matrix :
4 / 43
{
nucleotide
Sefuenus
.
:
4
otitis
.
=
A
,
C
,
G.
i
codons
i
64 Stites
or
61
Stites
amino
acids
:
20
Artes
morphological
acts
i
presence
/
absence
0
Itc
co
-
for
nucleotides
,
measured
in
expected
#
of
substitutions
per
site
⊥
first
-
order
-
mutation
probs
.
only
depend
on
the current
state
,
not
the
past
history
.
Q
-
-
¥
.
I -
t
)
a
-
-
int
:
:L
:S
:
the
next
instant
of
time

Markov models of substitution
I
For nucleotide sequence data, the most general rate matrix is given by
Q
c
c
a
[ A ] [ C ] [ G ] [ T ]
[ A ] ≠≠ μ a fi
C
μ b fi
G
μ c fi
T
[ C ] μ g fi
A
≠≠ μ d fi
G
μ e fi
T
[ G ] μ h fi
A
μ j fi
C
≠≠ μ f fi
T
[ T ] μ i fi
A
μ k fi
C
μ l fi
G
≠≠
R
d
d
b
where
I
μ = mean rate of substitution
I
fi
A
, fi
C
, fi
G
, fi
T
= base frequency parameters
I
a ≠ l are rates of specific types of changes
5 / 43
q
diagonal
ekoyenffg-yownyetirc.vn
Q
=
O
stationary
dirt
=
prob
.
of
A
(
e.g
.
)
at
a
sit
if
t
→
is

Markov models of substitution
I
Usually assume that the substitution process is time-reversible
I
Overall rate of change from base i to j is the same as from base j to i
I
Mathematically: fi
i
P
ij
( t ) = fi
j
P
ji
( t )
I
This forces some constraints on Q :
Q
c
c
a
[ A ] [ C ] [ G ] [ T ]
[ A ] ≠≠ μ a fi
C
μ b fi
G
μ c fi
T
[ C ] μ a fi
A
≠≠ μ d fi
G
μ e fi
T
[ G ] μ b fi
A
μ d fi
C
≠≠ μ f fi
T
[ T ] μ c fi
A
μ e fi
C
μ f fi
G
≠≠
R
d
d
b
I
This is called the General Time Reversible (GTR) model
6 / 43

Jukes-Cantor Model (JC69)
I
Simplest sub-model of GTR
I
Assumes that
I
all nucleotides are equally frequent
I
all mutations happen at the same rate
Q
c
c
a
[ A ] [ C ] [ G ] [ T ]
[ A ] ≠
3
4
μ
1
4
μ
1
4
μ
1
4
μ
[ C ]
1
4
μ ≠
3
4
μ
1
4
μ
1
4
μ
[ G ]
1
4
μ
1
4
μ ≠
3
4
μ
1
4
μ
[ T ]
1
4
μ
1
4
μ
1
4
μ ≠
3
4
μ
R
d
d
b
7 / 43
ITA
z
To
=
Tf
=
IT
,
=
LT
Q
-
-