Maximum Likelihood

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Maximum Likelihood
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Likelihood The likelihood is the probability of
the data given the model.
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Likelihood
If we flip a coin and get a head and we think the
coin is unbiased, then the probability of
observing this head is 0.5. If we think the coin
is biased so that we expect to get a head 80 of
the time, then the likelihood of observing this
datum (a head) is 0.8. The likelihood of making
some observation is entirely dependent on the
model that underlies our assumption. The datum
has not changed, our model has. Therefore under
the new model the likelihood of observing the
datum has changed.
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Maximum Likelihood (ML) ML assumes a explicit
model of sequence evolution. This is justifiable,
since molecular sequence data can be shown to
have arisen according to a stochastic
process. ML attempts to answer the question
What is the probability that I would observe
these data (a multiple sequence alignment) given
a particular model of evolution (a tree and a
process)?
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Likelihood calculations In molecular
phylogenetics, the data are an alignment of
sequences We optimize parameters and branch
lengths to get the maximum likelihood Each site
has a likelihood The total likelihood is the
product of the site likelihoods The maximum
likelihood tree is the tree topology that gives
the highest (optimized) likelihood under the
given model. We use reversible models, so the
position of the root does not matter.
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What is the probability of observing a G
nucleotide? If we have a DNA sequence of 1
nucleotide in length and the identity of this
nucleotide is G, what is the likelihood that we
would observe this G? In the same way as the
coin-flipping observation, the likelihood of
observing this G is dependent on the model of
sequence evolution that is thought to underlie
the data. Model 1 frequency of G 0.4
likelihood(G) 0.4 Model 2 frequency of G 0.1
likelihood(G) 0.1 Model 3 frequency of G
0.25 likelihood(G) 0.25
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What about longer sequences?

• If we consider a gene of length 2
• gene 1 GA
• The the probability of observing this gene is the
  product of the
• probabilities of observing each character
• Model frequency of G 0.4 frequencyof A
  0.15
• p(G) 0.4
  p(A) 0.15
• Likelihood (GA) 0.4 x 0.15 0.06

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or even longer sequences?

• gene 1 GACTAGCTAGACAGATACGAATTAC
• Model simple base frequency model
• p(A)0.15 p(C)0.2 p(G)0.4 p(T)0.25
  
• (the sum of all
  probabilities must equal 1)
• Likelihood (gene 1) 0.000000000000000018
  452813

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• Note about models
• You might notice that our model of base frequency
  is not the
• optimal model for our observed data.
• If we had used the following model
• p(A)0.4 p(C) 0.2 p(G) 0.2 p(T) 0.2
• The likelihood of observing the gene is
• L (gene 1) 0.000000000000335544320000
• L (gene 1) 0.000000000000000018452813

The datum has not changed, our model has.
Therefore under the new model the likelihood of
observing the datum has changed.
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Increase in model sophistication

• It is no longer possible to simply invoke a model
  that
• encompasses base composition, we must also
  include the
• mechanism of sequence change and stasis.
• There are two parts to this model - the tree and
  the process
• (the latter is confusingly referred to as the
  model, although
• both parts really compose the model).

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Different Branch Lengths

• For very short branch lengths, the probability of
  a character staying the
• same is high and the probability of it changing
  is low.
• For longer branch lengths, the probability of
  character change becomes
• higher and the probability of staying the same is
  lower.
• The previous calculations are based on the
  assumption that the branch
• length describes one Certain Evolutionary
  Distance or CED.
• If we want to consider a branch length that is
  twice as long (2 CED), then
• we can multiply the substitution matrix by itself
  (matrix2).

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Maximum Likelihood
Two trees each consisting of single branch
v 0.1
I (A) II (C)
v 1.0
I (A)
II (C)

• v mt
• m mutation rate
• t time

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Jukes-Cantor model
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I AACC II CACT
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1 j
N 1 C G G A C A C G T T T A C 2 C A G A C
A C C T C T A C 3 C G G A T A A G T T A A C 4
C G G A T A G C C T A G C
L(j) p
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Likelihood of the alignment at various branch
lengths
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Strengths of ML

• Does not try to make an observation of sequence
  change and then a correction for superimposed
  substitutions. There is no need to correct for
  anything, the models take care of superimposed
  substitutions.
• Accurate branch lengths.
• Each site has a likelihood.
• If the model is correct, we should retrieve the
  correct tree (If we have long-enough sequences
  and a sophisticated-enough model).
• You can use a model that fits the data.
• ML uses all the data (no selection of sites based
  on informativeness, all sites are informative).
• ML can not only tell you about the phylogeny of
  the sequences, but also the process of evolution
  that led to the observations of todays sequences.

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Weaknesses of ML

• Can be inconsistent if we use models that are not
  accurate.
• Model might not be sophisticated enough
• Very computationally-intensive. Might not be
  possible to examine all models (substitution
  matrices, tree topologies).

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Models

• You can use models that
• Deal with different transition/transversion
  ratios.
• Deal with unequal base composition.
• Deal with heterogeneity of rates across sites.
• Deal with heterogeneity of the substitution
  process (different rates
• across lineages, different rates at
  different parts of the tree).
• The more free parameters, the better your model
  fits your data (good).
• The more free parameters, the higher the variance
  of the estimate (bad).

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Choosing a Model Dont assume a model, rather
find a model that fits your data. Models often
have free parameters. These can be fixed to a
reasonable value, or estimated by ML. The more
free parameters, the better the fit (higher the
likelihood) of the model to the data.
(Good!) The more free parameters, the higher the
variance, and the less power to discriminate
among competing hypotheses. (Bad!) We do not
want to over-fit the model to the data
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What is the best way to fit a line (a model)
through these points?

• How to tell if adding (or removing) a certain
  parameter is a good idea?
• Use statistics
• The null hypothesis is that the presence or
  absence of the parameter makes no difference
• In order to assess signifcance you need a null
  distribution

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• We have some DNA data, and a tree. Evaluate the
  data with 3 different
• models.
• model ln likelihood ?
• JC -2348.68
• K2P -2256.73 91.95
• GTR -2254.94 1.79
• Evaluations with more complex models have higher
  likelihoods
• The K2P model has 1 more parameter than the JC
  model
• The GTR model has 4 more parameters than the K2P
  model
• Are the extra parameters worth adding?

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You can use the ?2 approximation to assess
significance of adding parameters
K2P vs GTR
JC vs K2P
We have generated many true null hypothesis data
sets and evaluated them under the JC model and
the K2P model. 95 of the differences are under
2.The statistic for our original data set was
91.95, and so it is highly significant. In this
case it is worthwhile to add the extra parameter
(tRatio). We have generated many true null
hypothesis data sets and evaluated them under the
K2P model and the GTR model. The statistic for
our original data set was 1.79, and so it is not
signifcant. In this case it is not worthwhile to
add the extra parameters.
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Bayesian Inference
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Maximum likelihood Search for tree that maximizes
the chance of seeing the data (P (Data
Tree)) Bayesian Inference Search for tree that
maximizes the chance of seeing the tree given the
data (P (Tree Data))
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Bayesian Phylogenetics Maximize the posterior
probability of a tree given the aligned DNA
sequences Two steps - Definition of the
posterior probabilities of trees (Bayes Rule) -
Approximation of the posterior probabilities of
trees Markov chain Monte Carlo (MCMC) methods
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Bayesian Inference
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Bayesian Inference
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Markov Chain Monte Carlo
Methods Posterior probabilities of trees are
complex joint probabilities that cannot be
calculated analytically. Instead, the posterior
probabilities of trees are approximated with
Markov Chain Monte Carlo (MCMC) methods that
sample trees from their posterior probability
distribution.
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MCMC A way of
sampling / touring a set of solutions,biased by
their likelihood 1 Make a random solution N1
the current solution 2 Pick another solution
N2 3 If Likelihood (N1 with N2 4 Else if Random (Likelihood (N2) /
Likelihood (N1)) then replace N1 with N2 5
Sample (record) the current solution 6 Repeat
from step 2
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Bayesian Inference
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Bayesian Inference