1. Recitation 5 2/4/09 ML in Phylogeny
Comp. Genomics
Recitation 5
2/4/09
ML in Phylogeny
Based on Slides by Ron Shamir and Nir Friedman

2. Outline Maximum likelihood (ML) ML in phylogeny
Ancestral sequence reconstruction using ML

3. Maximum likelihood One of the methods for parameter estimation
Likelihood: L=P(Data|Parameters)
Simple example:
Simple coin with P(head)=p
10 coin tosses
6 heads, 4 tails
L=P(Data|Params)=(106)p6 (1-p)4

4. Maximum likelihood We want to find p that maximizes L=(106)p6 (1-p)4
Infi 1, Remember?
Log is a monotolical function, we can optimize logL=log[(106)p6 (1-p)4]=
log(106)+6logp+4log(1-p)]
Deriving by p we get: 6/p-4/(1-p)=0
Estimate for p:0.6 (Makes sense?)

5. Likelihood of a Tree Input (small problem):
n sequences
A tree T, with labels on the leafs (X)
Find optimal labeled tree :
labeling of internal nodes (Y)
branch lengths (b)
Maximizing the likelihood P(X|T,Y,b)

6. Likelihood (2) How to compute P(X|T,Y,b)? Assumptions:
Each character is independent
The branching is a Markov process:
The probability of a node having a given label is only a function of the parent node and the branch length b between them.
The probabilities P(x|y,t) are known

7. Example x1 x2 x3 x4 x5 t1 t2 t3 t5

8. What if we want P(X|T,b)? Assume that the branch lengths b are known.
Independence
of sites
Markov property
independence of
each branch
ALGMB, December 01
© Ron Shamir , TAU

9. Properties of P Additivity: Reversibility
Allows to freely move the root

10. Efficient Likelihood Calculation (Felsenstein ’73)
Use dynamic prog. similar to parsimony
Need Sj(v,a) = Pr(subtree rooted in v | vj = x)
Initialization:
For each leaf v set Sj(v,a) = 1 if i is labeled by a, otherwise Sj(v,a) = 0
Recursion:
Traverse the tree in postorder: for each node v with children u and w, for each state x
Complexity: O(nmk2)
n species, m chars, k states

11. Ancestral sequence reconstruction
Input:
Rooted tree + extant (leaf) sequences
Substitution matrix + branch lengths
Problem: Find the sequence assignment of internal states which maximizes the total tree likelihood

12. Solving ancestral sequence reconstruction
Simple with parsimony methods, ≈ through the Fitch/Sankoff algorithms
Here, we’re interested in ML
Maximizing
P(ancestral S|contemporary S)
Joint vs. Marginal
Marginal: focus on a single node (e.g., the root), and maximize its likelihood
Joint: Infer all the sequences together

13. Solutions
We can enumerate all the possible ancestral states and check their likelihood…
cn possible combinations per character
n – number of internal nodes
Inapplicable when the tree is large
Koshi and Goldshtein (1996) – fast algorithm for marginal reconstruction
Pupko, Pe’er, Shamir and Graur (2004): fast algorithm for joint reconstruction

14. Basics We assume different sites evolve independently
Working one site at a time
Pij(t) – the probability of observing ij in time t
We want to maximize
P(v|data)=P(data|v)*P(v)/P(Data)
Constant!

15. DP to the rescue DP often suitable for tree problems Idea:
Start from the leaves and climb up the tree
The subtree under every node is dependent only on the state of its parent!
For node x compute Lx(i) and Cx(i)
Lx(i) – the likelihood of x’s subtree under the condition that its parent is assigned with i
Cx(i) – the state of x that gives rise to this likelihood

16. Algorithm phase I Initialization: Progression: Termination:
For a leaf y assigned with j: Cy(i)=j, Ly(i)=Pij(t)
Progression:
For an internal node z with sons x,y already visited: for each i we compute
Termination:
For the root with sons x,y,z – choose k maximizing

17. Algorithm phase II “Traceback”
Traverse the tree from root to the leaves
For every internal node x with father y already reconstructed with i
Reconstruct the state in x by setting Cx(i)
Continue until all the nodes are reconstructed

18. Example

19. Complexity
For n internal nodes and c possible states we compute a DP table of O(nc) cells.
As we maximize in every cell over c states, time is O(nc2)
As c is constant – O(n)