Likelihood Methods

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Likelihood Methods
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Likelihood maximization is a very common approach
to inference
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A coin is known to be biased The coin is tossed
three times two heads and one tail Use
principal of ML to estimate the probability of
throwing a head
Try P 0.2
L(D) 0.2 0.2 0.8 0.032
Try P 0.4
L(D) 0.4 0.4 0.6 0.096
Likelihood of the Data
Try P 0.6
L(D) 0.6 0.6 0.4 0.144
Try P 0.8
L(D) 0.8 0.8 0.2 0.128
Probability of a head
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Genetic Distance
Given a model of the way sequences evolve we can
derive a function that describes the likelihood
of observing a particular pair of sequences as a
function of the inferred genetic distance between
them
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A C G G A G Consider simplified Jukes and Cantor
model
D 0.3 0.3 0.3 0.7 0.063
D 0.6 0.6 0.6 0.4 0.144
Likelihood
D 0.9 0.9 0.9 0.1 0.081
Note this is a major simplification
Distance
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Genetic Distance using Maximum Likelihood

• Require a model of evolution
• Optimise all parameters of the model
• Each evolutionary event has an associated
  likelihood given an inferred genetic distance
• The likelihood of the sequence-pair is a function
  of the genetic distance (just the product of the
  likelihoods of each of the inferred events at
  each sequence position)
• Function is minimized

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Phylogenetic trees using Maximum Likelihood

• Require a model of evolution
• Each substitution has an associated likelihood
  given a branch of a certain length
• A function is derived to represent the likelihood
  of the data given the tree, branch-lengths and
  additional parameters
• Optimise over parameters of the model
• Optimise over branch lengths
• Sum the likelihood over all possible sequences at
  ancestral nodes
• Search for the best tree (using heuristics such
  as TBR)

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Models can be made more parameter rich to
increase their realism

• The most common additional parameters are
• A correction to allow different rates for each
  type of nucleotide change
• A correction for the proportion of sites which
  are unable to change
• A correction for variable rates at those sites
  which can change
• The values of the additional parameters will be
  estimated in the process

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A gamma distribution can be used to model site
rate heterogeneity
Can be iterative.

• Rates of evolution can be worked out accurately
  if the tree is known
• Accurate rates of evolution for each sequence
  position improve the accuracy of the tree

Rates programme
Rates
Tree
Tree programme
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Likelihood and the number of parameters

• More parameters always leads to a better fit of
  the data

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More parameters always leads to a higher value of
the likelihood whether or not the additional
parameters are providing a significantly better
fit to the data
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• Are the extra parameters justified?
• - Likelihood ratio test (applies to nested
  models)
• - Akaike Information Criterion

dof number of additional parameters
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One model is nested in another if it is a special
case of the more general model e.g. the Jukes
and Cantor model and Kimura 2P model
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• Modeltest
• - Uses PAUP
• - Tries out many nested models of nucleotide
  substitution
• - Decides how many parameters are justified by
  the data

GTR does not overfit the data for at least some
HIV sequences
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Likelihood-based tests of topologies

• Kishino-Hasegawa test
• Trees specified apriori
• KH can be used to test whether two competing
  hypotheses have significantly different
  likelihood
• Involves non-parametric bootstrap to get an idea
  of by how much the likelihoods vary
• NB should not be used to test trees that have
  been chosen on the basis of the data!

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Likelihood-based tests of topologies

• Shimodaira-Hasegawa test
• Can be used to test confidence of ML tree
  compared to related trees (e.g. second most
  likely tree from the data)
• PAUP, Andrew Rambaut http//evolve.zoo.ox.ac.uk/so
  ftware/shtests

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Inferring Sequences at Ancestral Nodes

• Maximum likelihood estimates of tree topologies
  also provide inferred sequences at ancestral
  nodes
• Analysis of sequences at ancestral nodes and
  sequence changes at ancestral branches can
  provide information about the timing of the
  acquiring of a novel trait or mutation
• PAML (Phylogenetic Analysis using Maximum
  Likelihood)
• Confidence intervals provided
• Selection can be inferred