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Maximum Likelihood

Flips usage of probability function A typical calculation: P(h|n,p) = C(h, n) * p h * (1-p) (n-h) The implied question: Given p of success in single trial, what is probability of h success over n trials?. Maximum Likelihood. The ML question. The ML calculation:

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Maximum Likelihood

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  1. Flips usage of probability function A typical calculation: P(h|n,p) = C(h, n) * ph * (1-p)(n-h) The implied question: Given p of success in single trial, what is probability of h success over n trials? Maximum Likelihood

  2. The ML question The ML calculation: L(p|nh) = C(h, n) * ph * (1-p)(n-h) What is probability that parameter p results in h success over n trials? Experiment with test values of p and choose the one that results in highest likelihood

  3. AG AC TG Let #sequences = s = 3 Each position a data point For each position, 4s possible values, e.g. {A,A,A},{A,A,T}… In this example, 64 possible values each position. Consider a small alignment

  4. Probability/Likelihood function The simplest model – use an arbitrary p for each of the 64 possible values based on its observed freq. 2 patterns have p=0.5, all others p=0. Result “works” but is not biologically interesting.

  5. Maximum likelihood testing model

  6. Definition • Method for the inference of phylogeny • Method that searches for the tree with the highest probability or likelihood.

  7. Example going through the Maximum likelihood model • Assume that we have the aligned nucleotide sequences for four taxa: (1) A G G C U C C A A ....A (2) A G G U U C G A A ....A (3) A G C C C A G A A.... A • A U U U C G G A A.... C Evaluate the likelihood of the uprooted tree represented by the nucleotides of site j in the sequence http://www.icp.ucl.ac.be/~opperd/private/max_likeli.html

  8. Since the likelihood of the tree is independent of the position of the root, we can display the figure as shown in Figure B. • Assume that the nucleotides evolve independently (the Markovian model of evolution) • Calculate the likelihood for each site separately and combine the likelihood into a total value towards the end. • . To calculate the likelihood for site j, we have to consider all the possible scenarios by which the nucleotides present at the tips of the tree could have evolved. • Therefore the likelihood for a particular site is the summation of the probabilities of every possible reconstruction of ancestral states, given some model of base substitution. http://www.icp.ucl.ac.be/~opperd/private/max_likeli.html

  9. So in this specific case all possible nucleotides A, G, C, and T occupying nodes (5) and (6), or 4 x 4 = 16 possibilities: • Protein sequences each site may occupy 20 states (that of the 20 amino acids) • 20x20 thus 400 possibilities have to be considered. • Since any one of these scenarios could have led to the nucleotide configuration at the tip of the tree, we must calculate the probability of each and sum them to obtain the total probability for each site j. http://www.icp.ucl.ac.be/~opperd/private/max_likeli.html

  10. The likelihood for the full tree then is product of the likelihood at each site. • Since the individual likelihoods are extremely small numbers it is convenient to sum the log likelihoods at each site and report the likelihood of the entire tree as the log likelihood.

  11. This above procedure is then repeated for all possible topologies (for all possible trees). • The tree with the highest probability is the tree with the highest maximum likelihood.

  12. Hulsenbeck J., Crandall, K. Annu. Rev. Ecol. Syst., 1997, 28:437-66.

  13. DNA Substitution Models

  14. General DNA Substitution Model Likelihood L is the propability of observing data D given hypothesis H L = Pr(D/H) The use of maximum likelihood (ML) algorithms in developing phylogenetic hypotheses requires a model of evolution.

  15. The rate matrix for a general model of DNA substitution is given by The rows and columns are ordered A, C, G and T. The matrix gives the rate of change from nucleotide i(arranged along the rows) to nucleotide j(along the columns). For example r2pC gives the rate of change from A to C.

  16. Let P(v,s) be the transition probability matrix where pi,j(v,s) is the probability that nucleotide i changes into j over branch length v. The vector s contains the parameters of the substitution model(eg. pA, pC, pG, pT, r1,r2…). For two-state case, to calculate the probability of observing a change over a branch of length v, the following matrix calculation is performed: P (v,s) = eQv

  17. DNA substitution Models

  18. Advantages of Maximum likelihood • Lower variance than other methods • Least affected by sampling error • Robust to many violations of the assumptions of the evolutionary model, even with very short sequences, they outperform other methods). • Are less error prone. • Statistically well founded. • Evaluate different tree topologies.

  19. Disadvantages of Maximum likelihood • CPU intensive and may take a long time to complete an evaluation • The result is dependent on the model of evolution used.

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