Maximum likelihood

Maximum likelihood

The maximum likelihood criterion

• The optimal tree is that which would be most likely to give rise to the observed data (under a given model of evolution)

An outline of the ML approach:Consider one character, i

(It is useful to arbitrarily root the tree)

Sum across all possible histories for i

There are 4(n-2) arrangements for n taxa

We calculate the likelihood of getting the observed states = L(i)

A GGG

A

A

t2 t3 t4 t5

t1

L(i) = PA x PA-A(t1)x PA-G(t2)x PA-G(t3)x PA-A(t4)x PA-G(t5)

Multiply across all sites (assume independence)

L will be very small(lnL will be a large negative number)

Tree searching

• Search for the set of branch-lengths that maximize L (= lower -lnL score)

• Record that score

• Search for tree topologies with the best score

Time consuming!

Issues glossed over

• Where do we get Pn - the probability of state n at the arbitrary root node?– Equiprobable (25%)– Empirical (frequency in the entire matrix)– Estimated (optimized by ML on each tree)

• Where do we get Pi-j(t) - the probability of going from state i to state j in time t?

Typical Simplifying Assumptions

• Stationarity

• Reversibility

• Site independence

• Markovian process (no “memory”)

The simplest model of molecular evolution: Jukes-Cantor

A C G T

A -3

C -3

G -3

T -3

Instantaneous rate matrix (Q-matrix)

Calculating probabilities of change

• To convert the Q matrix into a matrix giving the probability of starting at state i and ending in state j, t time units later uses the formula:

P(t)= eQt

The simplest model of molecular evolution: Jukes-Cantor

A C G T

A

C

G

T

Substitution probability matrix (P-matrix)

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A A

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A A

Non-change =¼ +¾e-t

Change = ¼ - ¼e-t

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A A

L =0.25(¼ +¾e-0.02) (¼ +¾e-0.02) (¼ +¾e-0.03) (¼ - ¼e-0.01) (¼ - ¼e-0.01)

Non-change =¼ +¾e-t

Change = ¼ - ¼e-t

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A A

L =(¼ +¾e-0.02) (¼ +¾e-0.02) (¼ +¾e-0.03) (¼ - ¼e-0.01) (¼ - ¼e-0.01)

Non-change =¼ +¾e-t

Change = ¼ - ¼e-t

L =(0.985149) (0.985149) (0.977834) (0.0024875) (0.0024875)

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A A

L =(¼ +¾e-0.02) (¼ +¾e-0.02) (¼ +¾e-0.03) (¼ - ¼e-0.01) (¼ - ¼e-0.01)

Non-change =¼ +¾e-t

Change = ¼ - ¼e-t

L =(0.985149) (0.985149) (0.977834) (0.0024875) (0.0024875)L = 0.00000587232 or 5.87232 x 10-6

Worked example

0.03

0.01

0.01

0.02

0.02

A

A G

G

A G

L =(¼ +¾e-0.02) (¼ +¾e-0.02) (¼ - ¼e-0.03) (¼ +¾e-0.01) ¼ +¾e-0.01)

Non-change =¼ +¾e-t

Change = ¼ - ¼e-t

L =(0.985149) (0.985149) (0.0073886) (0.9925373) (0.9925373)L = 0.007064163 or 7.064163 x 10-3

Sum over all other combinations

• AA = 5.87232x10-6

• AG = 7.064163x10-3

• AC =

• AT =

• GA =

• GG =

• GC =

• GT =

• CA =

• CG =

• CC =

• CT =

• TA =

• TG =

• TC =

• TT =

Sum over all other combinations

• AA = NC-NC-C

• AG = NC-C-NC

• AC = NC-C-C

• AT = NC-C-C

• GA = C-C-C

• GG = C-NC-NC

• GC = C-C-C

• GT = C-C-C

• CA = C-C-C

• CG = C-C-NC

• CC = C-NC-C

• CT = C-C-C

• TA = C-C-C

• TG = C-C-NC

• TC = C-C-C

• TT = C-NC-C

Sum over all other combinations

• AA = 5.87232x10-6

• AG = 7.064163x10-3

• AC = 4.43719x10-8

• AT = 4.43719x10-8 • GA = 1.1204x10-12

• GG = 2.36063x10-5 • GC = 1.1204x10-12

• GT = 1.1204x10-12

• CA = 1.1204x10-12

• CG = 1.78372x10-7 • CC = 1.48277x10-12

• CT = 1.1204x10-12

• TA = 1.1204x10-12

• TG = 1.78372x10-7

• TC = 1.1204x10-12

• TT = 1.48277x10-10

Sum over all other combinations = 7.09 x 10-3

• AA = 5.87232x10-6

• AG = 7.064163x10-3

• AC = 4.43719x10-8

• AT = 4.43719x10-8 • GA = 1.1204x10-12

• GG = 2.36063x10-5 • GC = 1.1204x10-12

• GT = 1.1204x10-12

• CA = 1.1204x10-12

• CG = 1.78372x10-7 • CC = 1.48277x10-12

• CT = 1.1204x10-12

• TA = 1.1204x10-12

• TG = 1.78372x10-7

• TC = 1.1204x10-12

• TT = 1.48277x10-10

Likelihood scores

• Raw likelihood of the data at this site given this tree and branch lengths and model = 0.25(7.09 x 10-3)

• Log-likelihood = -6.334787983

What does this number mean?

• -6.334787983 = The log-likelihood of the data (tip values) given:– This tree topology– These branch lengths– The model of molecular evolution

Multiplying across sites

To make it easier, we can lump characters with the same “pattern”lnL = [lnL (0000)]N(0000) +[lnL (0001)]N(0001) + [lnL (0010)]N(0010) + [lnL (0100)]N(0100) + [lnL (0111)]N(0111) + [lnL (0011)]N(0011) + [lnL (0101]N(0101)

) + [lnL (0110)]N(0110)

What branch lengths should we assume?

• Under the principle of maximum likelihood, we use the set of branch lengths that maximize the likelihood

• Once we find those branch lengths, the likelihood score is taken as being the likelihood of the data given this tree topology