# Phylogenetics: Likelihood

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1 1 Phylogenetics: Likelihood COMP 571 Luay Nakhleh, Rice University The Problem 2 Input: Multiple alignment of a set S of sequences Output: Tree T leaf-labeled with S Assumptions 3 Characters are mutually independent Following a speciation event, characters continue to evolve independently

2 4 The likelihood of model M given data D, denoted by L(M D), is p(d M). For example, consider the following data D that result from tossing a coin 10 times: HTTTTHTTTT 5 Model M1: A fair coin (p(h)=p(t)=0.5) L(M1 D)=p(D M1)= Model M2: A biased coin (p(h)=0.8,p(t)=0.2) L(M2 D)=p(D M2)=

3 7 Model M3: A biased coin (p(h)=0.1,p(t)=0.9) L(M3 D)=p(D M3)= The problem of interest is to infer the model M from the (observed) data D. 9 The maximum likelihood estimate, or MLE, is: ˆM argmax M p(d M)

4 10 D=HTTTTHTTTT M1: p(h)=p(t)=0.5 M2: p(h)=0.8, p(t)=0.2 M3: p(h)=0.1, p(t)=0.9 MLE (among the three models) is M3. 11 A more complex example: The model M is an HMM The data D is a sequence of observations Baum-Welch is an algorithm for obtaining the MLE M from the data D 12 The model parameters that we seek to learn can vary for the same data and model. For example, in the case of HMMs: The parameters are the states, the transition and emission probabilities (no parameter values in the model are known) The parameters are the transition and emission probabilities (the states are known) The parameters are the transition probabilities (the states and emission probabilities are known)

5 Back to Phylogenetic Trees 13 What are the data D? A multiple sequence alignment (or, a matrix of taxa/characters) Back to Phylogenetic Trees 14 What is the (generative) model M? The tree topology The branch lengths The model of evolution (JC,..) Back to Phylogenetic Trees 15 What is the (generative) model M? The tree topology, T The branch lengths, λ The model of evolution (JC,..), Ε

6 Back to Phylogenetic Trees 16 The likelihood is p(d T,λ,Ε). The MLE is ( ˆT,ˆ, Ê) argmax (T,,E)p(D T,,E) Back to Phylogenetic Trees 17 In practice, the model of evolution is estimated from the data first, and in the phylogenetic inference it is assumed to be known. In this case, given D and E, the MLE is ( ˆT,ˆ) argmax (T, ) p(d T, ) Assumptions 18 Characters are independent Markov process: probability of a node having a given label depends only on the label of the parent node and branch length between them t

7 Maximum Likelihood 19 Input: a matrix D of taxa-characters Output: tree T leaf-labeled by the set of taxa, and with branch lengths λ so as to maximize the likelihood P(D T,λ) P(D T,λ) 20 P (D T, ) = Q site j p(d j T, ) = Q site j (P R p(d j,r T, )) = Q P site j R hp(root) Q i edge u!v p u!v(t uv ) 21 What is pi j(tuv) for a branch u v in the tree, where i and j are the states of the site at nodes u and v, respectively?

8 22 For the Jukes-Cantor model with the parameter μ (the overall substitution rate), we have 1 p i!j (t) = 4 (1 + 3e tµ ) i = j 1 4 (1 e tµ ) i 6= j 23 If branch lengths are measured in expected number of mutations per site, ν (for JC: ν=(μ/4+μ/4+μ/4)t=(3/4)μt) 1 p i!j ( ) = 4 (1 + 3e 4 /3 ) i = j 1 4 (1 e 4 /3 ) i 6= j 24 The ML problem is NP-hard (that is, finding the MLE (T,λ) is very hard computationally) Heuristics involve searching the tree space, while computing the likelihood of trees Computing the likelihood of a leaf-labeled tree T with branch lengths can be done efficiently using dynamic programming

9 P(D T,λ) Let Cj(x,v) = P(subtree whose root is v vj=x) Initialization: leaf v and state x C j(x, v) = { 1 vj = x 0 otherwise Recursion: node v with children u,w [ ] [ ] Cj(x, v) = Cj(y, u) Px y(tvu) Cj(y, w) Px y(tvw) y y 25 Te rmin at ion: [ ] m L = Cj(x, root) P(x) j=1 x Running Time 26 Takes time O(nk 2 m), where n is the number of leaves in the tree, m is the number of sites, and k is the maximum number of states per site (for DNA, k=4) Unidentifiability of the Root 27 If the base substitution model is reversible (most of them are!), then rooting the same tree differently doesn t change the likelihood.

10 28 Questions?

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