Presentation taken from Nir Friedman’s HU course, available at cs.huji.ac.il/~pmai .

1. Maximum Likelihood (ML) Parameter Estimation with applications to reconstructing phylogenetic treesComput. Genomics, lecture 6b Presentation taken from Nir Friedman’s HU course, available at www.cs.huji.ac.il/~pmai. Changes made by Dan Geiger, Ydo Wexler, and finally by Benny Chor. .

2. The Setting • We have a probabilistic model,M, of some phenomena. We know exactly the structure of M, but not the values of its probabilistic parameters, . • Each “execution” of M produces an observation, x[i] , according to the (unknown) distribution induced by M. • Goal: After observing x[1] ,…, x[n] , estimate the model parameters, , that generated the observed data.

3. Maximum Likelihood Estimation (MLE) • The likelihood of the observed data, given the model parameters, as the conditional probabilitythat the model, M, with parameters , produces x[1] ,…, x[n] . L()=Pr(x[1] ,…, x[n] | , M), • In MLE we seek the model parameters, , that maximize the likelihood.

4. Maximum Likelihood Estimation (MLE) • In MLE we seek the model parameters, , that maximize the likelihood. • The MLE principle is applicable in a wide variety of applications, from speech recognition, through natural language processing, to computational biology. • We will start with the simplest example: Estimating the bias of a coin. Then apply MLE to inferring phylogenetic trees. • (will later talk about MAP - Bayesian inference).

5. Example: Binomial Experiment • When tossed, it can land in one of two positions: Head(H) or Tail (T) Head Tail • We denote by  the (unknown) probability P(H). Estimation task: • Given a sequence of toss samples x[1], x[2], …, x[M] we want to estimate the probabilities P(H)= and P(T) = 1 - 

6. i.i.d. Samples (why??) Statistical Parameter Fitting (restement) • Consider instances x[1], x[2], …, x[M] such that • The set of values that x can take is known • Each is sampled from the same distribution • Each sampled independently of the rest • The task is to find a vector of parameters  that have generated the given data. This vector parameter  can be used to predict future data.

7. L()  0 0.2 0.4 0.6 0.8 1 The Likelihood Function • How good is a particular ?It depends on how likely it is to generate the observed data • The likelihood for the sequence H,T, T, H, H is

8. Sufficient Statistics • To compute the likelihood in the thumbtack example we only require NH and NT (the number of heads and the number of tails) • NH and NT are sufficient statistics for the binomial distribution

9. Formally, s(D) is a sufficient statistics if for any two datasets D and D’ • s(D) = s(D’ ) LD() = LD’ () Datasets Statistics Sufficient Statistics • A sufficient statistic is a function of the data that summarizes the relevant information for the likelihood

10. Maximum Likelihood Estimation MLE Principle: Choose parameters that maximize the likelihood function • This is one of the most commonly used estimators in statistics • Intuitively appealing • One usually maximizes the log-likelihood function, defined as lD() = ln LD()

11. L() 0 0.2 0.4 0.6 0.8 1 Example: (NH,NT ) = (3,2) MLE estimate is 3/5 = 0.6 Example: MLE in Binomial Data Taking derivative and equating it to 0, we get (which coincides with what one would expect)

12. Sufficient statistics: • N1, N2, …, NK - the number of times each outcome is observed Likelihood function: MLE: (proof @ assignment 3) From Binomial to Multinomial • Now suppose X can have the values 1,2,…,K(For example a die has K=6 sides) • We want to learn the parameters 1, 2. …, K

13. MLE: Example: Multinomial • Let be a protein sequence • We want to learn the parameters q1, q2,…,q20 corresponding to the frequencies of the 20 amino acids • N1, N2, …, N20 - the number of times each amino acid is observed in the sequence Likelihood function:

14. Inferring Phylogenetic Trees • Let be n sequence(DNA or AA). Assume for simplicity they are all same length, l. • We want to learn the parameters of a phylogenetic tree that maximizes the likelihood. • But wait: Should first specify a model.

15. A Probabilistic Model • Our models will consist of a “regular” tree, where in addition, edges are assigned substituion probabilities. • For simplicity, assume our “DNA” has only two states, say X and Y. • If edge eis assigned probability pe, this means that the probability of substitution (X Y) across e is pe.

16. A Probabilistic Model (2) • Our models will consist of a “regular” tree, where in addition, edges are assigned substituion probabilities. • For simplicity, assume our “DNA” has only two states, say X and Y. • If edge eis assigned probability pe, this means that the probability of substitution (X Y) across e is pe.

17. A Probabilistic Model (3) • If edge eis assigned probability pe, this means that the probability of more involved patterns of substitution across e(e.g.XXYXYYXYXX) is determined, and easily computed: pe2(1- pe)3 for this pattern. • Q.: What if pattern on both sides is known, but pe is not known? • A.: Makes sense to seek pe that maximizes probability of observation. • So far, this is identical to coin toss example.

18. A Probabilistic Model (4) But a single edge is a fairly boring tree… Now we don’t know the states at internal node(s), nor the edge parameters pe1, pe2, pe3 YXYXX XXYXY pe2 pe1 pe3 ????? YYYYX

19. Two Ways to Go 1. Maximize over states of internal node(s) 2. Average over states of internal node(s) In both cases, we maximize over edge parameters YXYXX XXYXY pe2 pe1 pe3 ????? YYYYX

20. Two Ways to Go In the first version (average, or sum over states of internal nodes) we are looking for the “most likely” setting of tree edges. This is called maximum likelihood (ML) inference of phylogenetic trees. ML is probably the inference method most widely (some would say wildly ) used. YXYXX XXYXY pe2 pe1 pe3 ????? YYYYX

21. Two Ways to Go YXYXX XXYXY pe2 In the second version (maximize over states of internal nodes) we are looking for the “most likely” ancestral states. This is called ancestral maximum likelihood (AML). In some sense AML is “between” MP (having ancestral states) and ML (because the goal is still to maximize likelihood). pe1 pe3 ????? YYYYX

22. Back to the Probabilistic Model pe1 pe3 A reconstruction method is called statistically consistent if the model it reconstructs converges to the “true tree”as thelengthof the sequences goes to infinity. XXYXY YXYXY pe4 pe2 pe5 YYYYX XXXYX

23. Consistency, and Beyond A reconstruction method is called statistically consistent if the model it reconstructs converges to the “true tree”as the length of the sequences goes to infinity. We would like a reconstruction method that is • Statistically Consistent • Computationally efficient

24. Status Report Let us examine the three character based methods we saw in light of these two criteria: Statistical consistency & Computational efficiency