1. Maximum Likelihood
Given competing explanations for a particular observation, which explanation should we choose?
Maximum likelihood methodologies suggest that the explanation that makes the observation most likely is the preferred one
Given some data, D, and a hypothesis, H,
LD = Pr(D|H) - the likelihood of the data is the probability of observing D given H
Previous slide = L = P(D | M, θ, τ, ν)
For our purposes, D is the data set (sequences typically) and H is any possible tree relating those sequences
The best tree is the one that makes the observed data most likely.
The main idea behind maximum likelihood (ML) phylogenetic inference is to determine the tree topology, branch lengths, and evolutionary model that maximizes the probability of observing the sequences observed
L(τ, θ) = Prob(Data | τ, θ) = Prob(Aligned sequences | tree, model)

2. Maximum Likelihood Given four taxa and associated sequences
1 – …TTCGCTTAA…
2 – …TTTCTGCAA…
3 – …TTGCTGGTA…
4 – …TCTCGGCAA…
If we have an evolutionary model, we have an estimate of the instantaneous rates of change for any given site and set of nucleotides
We can also derive any number of hypothetical trees on which to map the data
Our job is to determine the likelihood of data given the evolutionary model and the possible trees

3. Maximum Likelihood Given four taxa and associated sequences
1 – …TTCGCTTAA…
2 – …TTTCTGCAA…
3 – …TTGCTGGTA…
4 – …TCTCGGCAA…
For the bold position, there are three possible trees
Assuming a given evolutionary model, a value can be assigned to each topology C C C G
T(2) G Y Y Y X
T(3) X G X
T(2)
T(2)
T(3)
T(3)

4. Maximum Likelihood
1 – …TTCGCTTAA…
2 – …TTTCTGCAA…
3 – …TTGCTGGTA…
4 – …TCTCGGCAA…
Just as we are considering only one site among many, we can consider one tree among many
This is one possible rooted tree
There are 16 possible values for X and Y
Again, let’s choose one of them G C
T(2)
T(3) Y T X A

5. Maximum Likelihood
1 – …TTCGCTTAA…
2 – …TTTCTGCAA…
3 – …TTGCTGGTA…
4 – …TCTCGGCAA…
Since we have a model of evolutionary change we can calculate the probability of this tree for this site
It is a product of the probability of all of the states/changes of state given our model of sequence evolution
P(τ) = Π Pi, a product function
P(τ) = PA x PAG x PAC x PAT x PTT x PTT N
i=1 G C
T(2)
T(3) T A

6. Maximum Likelihood
1 – …TTCGCTTAA…
2 – …TTTCTGCAA…
3 – …TTGCTGGTA…
4 – …TCTCGGCAA…
The probability must be calculated for all sites for this tree
P(τ) = Π Pi, a product function
Then for all sites in all possible trees
These numbers are very small so they are typically expressed as log likelihoods
ln L(τ) = Σ lnLi
ln L(τ) is the log likelihood of observing the given alignment under the chosen evolutionary model, given that particular tree and branch lengths on the tree
Because we are dealing not only with simple tree topologies but also with branch lengths, there are even more trees than ordinarily considered
Heuristic (approximate) methods are usually applied N
i=1 N G C
T(2)
T(3)
i=1 T A

7. Maximum Likelihood
Because we are dealing not only with simple tree topologies but also with branch lengths, there are even more trees than ordinarily considered
To reduce the computational complexity, heuristic methods are usually applied to suggest reasonable starting trees
Exact methods – will find the best tree under a given criterion but not feasible for large data sets
Branch and Bound
Heuristic - any approach to problem solving, learning, or discovery that employs a practical methodology not guaranteed to be optimal or perfect, but sufficient for the immediate goal
Stepwise addition
Branch swapping methods
Quartet puzzling

8. Maximum Likelihood Branch-and-Bound Method Good for 12 – 25 taxa
Add taxa to trees along ‘paths’
Quit a path when it is apparent that no solutions along that path are optimal
Accomplished by evaluating tree criterion after each addition
Good for 12 – 25 taxa
Will find a locally optimal tree

9. Maximum Likelihood Branch-and-Bound Method L = number of terminal taxa
Choose an initial tree with three leaves from L
Add a terminal taxon at a defined position
Repeat until all taxa are added
Evaluate using optimality criterion
Set upper bound for optimality criterion
Repeat

10. Maximum Likelihood Branch-and-Bound Method Taxa A-F evaluated in
L = number of terminal taxa
Choose an initial tree with three leaves from L
Add a terminal taxon at a defined position
Repeat until all taxa are added
Evaluate using optimality criterion
Set upper bound for optimality criterion
Repeat
Taxa A-F evaluated in
this example

11. Maximum Likelihood Stepwise Addition Method
Select three random taxa from n terminal taxa
Find the most likely tree
Add another random taxon
Repeat n-3 times
Will find a locally optimal tree
Other addition orders may give a more optimal tree
Perform tree rearrangements to search for other optimal trees

12. Maximum Likelihood Stepwise Addition Method
Select three random taxa from n terminal taxa
Find the most likely tree
Add another random taxon
Repeat n-3 times
Will find a locally optimal tree
Other addition orders may give a more optimal tree
Perform tree rearrangements to search for other optimal trees

13. Maximum Likelihood
Once you have found a reasonable tree using a heuristic method…
Perform branch swapping to search for other, possibly more optimal trees
Nearest neighbor interchange (NNI)
Subtree pruning and regrafting (SPR)
Tree bisection and reconnection (TBR)
NNI
TBR
SPR

14. Maximum Likelihood
Nearest neighbor interchange (NNI)
For any internal edge, there are three ways the four subtrees can be regrouped

15. Maximum Likelihood
Subtree pruning and regrafting (SPR)
Clip subtrees and reinsert them at all possible locations

16. Maximum Likelihood Cut a tree into two subtrees
Tree bisection and reconnection (TBR)
Cut a tree into two subtrees
Reconnect the trees by creating a new branch that joins one subtree to a branch on the other

17. Maximum Likelihood Quartet Puzzling Method
Given any set of sequences, any group of four is a quartet

18. Maximum Likelihood Quartet Puzzling Method
1. Estimate parameters for the model to be used
Build distance matrix (D) and corresponding NJ tree using a given model
Determine ML branch lengths and use to re-estimate model parameters
Using new estimates, rebuild D and NJ tree, re-estimate parameters
Iterate second two steps until the parameters are stable
2. Calculate likelihoods for all quartets = 3 x (n!/(4!(n-4)!))
3. Add taxa in random order and positioned in least contradictory position based on likelihoods
Repeat using different addition orders to generate a set of trees
4. Build a consensus tree where the percent occurrence of each branch is represented with puzzle support values

19. Maximum Likelihood Quartet Puzzling Method Assume 6 taxa
1. Pick four at random and build the best ML tree
2. Pick another random sequence and add it for all possible quartets
3. Evaluate ML for each
4. Graft new taxon on best branch based on ML
5. Repeat
2,3
best 4 4’
2’,3’
best

20. Maximum Likelihood
All of these methods will work for ML, parsimony, Bayesian methods

21. Maximum Likelihood Objections to ML
Phylogenetic inferences using ML require an explicit model of evolution
Good – we are aware of any assumptions
Bad – where do we get our parameter estimates?
If we knew the actual parameters, we could better infer evolution
In order to get the actual parameters, we need to know the evolutionary history