Maximum Likelihood

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Maximum Likelihood

• Flips usage of probability function
• A typical calculation
• P(hn,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?

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The ML question

• The ML calculation
• L(pnh) 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

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Consider a small alignment

• 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.

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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 p0.5, all others
p0. Result works but is not biologically
interesting.
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Maximum likelihood testing model
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Definition

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

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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_likel
i.html
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• 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_likel
i.html
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• 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_likel
i.html
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• 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.

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• 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.

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Hulsenbeck J., Crandall, K. Annu. Rev. Ecol.
Syst., 1997, 28437-66.
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DNA Substitution Models
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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.
•  

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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.
 
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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
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DNA substitution Models
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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.

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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.