Maximum likelihood

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Maximum likelihood
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The maximum likelihood criterion

• The optimal tree is that which would be most
  likely to give rise to the observed data (under a
  given model of evolution)

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How it differs from parsimony (from Swofford et
al. 1996)

• What can we say about the placement of another
  taxon with state C?

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How it differs from parsimony (from Swofford et
al. 1996)

• Parsimony the new taxon could attach in several
  places

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How it differs from parsimony (from Swofford et
al. 1996)

• ML - One place is favored
• State at ? most likely A

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An outline of the ML approachConsider one
character, i
(It is useful to arbitrarily root the tree)
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Sum across all possible histories for i
There are 4(n-2) arrangements for n taxa
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For each tree we calculate the likelihood of
getting the observed states L(i)
A
G
G
G
t2
t3
t4
t5
A
t1
A
L(i) PA x PA-A(t1)x PA-G(t2)x PA-G(t3)x
PA-A(t4)x PA-G(t5)
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Multiply across all sites (assume independence)
L will be very small(lnL will be a large
negative number)
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Tree searching

• Search for the set of branch-lengths that
  maximize L ( lower -lnL score)
• Record that score
• Search for tree topologies with the best score

Time consuming
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Issues glossed over

• Where do we get Pn - the probability of state n
  at the arbitrary root node?
• Equiprobable (25)
• Empirical (frequency in the entire matrix)
• Estimated (optimized by ML on each tree)
• Where do we get Pi-j(t) - the probability of
  going from state i to state j in time t?

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Typical Simplifying Assumptions

• Stationarity
• Reversibility
• Site independence
• Markovian process (no memory)

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The simplest model of molecular evolution
Jukes-Cantor
Instantaneous rate matrix (Q-matrix)
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Calculating probabilities of change

• To convert the Q matrix into a matrix giving the
  probability of starting at state i and ending in
  state j, t time units later uses the formula

P(t) eQt
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The simplest model of molecular evolution
Jukes-Cantor
Substitution probability matrix (P-matrix)
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More complicated (realistic) models for DNA

• Allow deviation from equiprobable base
  frequencies
• HKY85 F81GTR
• Allow two substitution types (ti and tv)
• K2P HKY85
• Allow for six substitution types
• GTR

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Relationship among models
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Relationship between MP and ML

• One argument - MP is inherently nonparametric ?
  No direct comparison possible
• MP is an ML model that makes particular
  assumptions

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The Goldman (1990) model(see Lewis 1998 for more)

• We force all branch lengths to be equal
• The Likelihood for a character only includes the
  set of ancestral states that maximizes the
  likelihood

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Why use MP

• The model is clearly less realistic, but
• We can do more thorough searches and data
  exploration (computational efficiency)
• Robust results will usually still be supported

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Why use ML

• The model (assumptions) are explicit
• We can statistically compare alternative models
• We can conduct parametric statistical tests
  (under the assumption that we have used the
  correct model)
• But, even the most complex model is still
  unrealistically simple