Model Selection and Model Averaging in Phylogenetics
Johan Nylander
November 27, 2005
1
Key Concepts
•
Model selection  to use the data to select a model  should be an integral part of inference
[4].
•
The data generating or ”true” model (
f
) has an inFnite number of parameters and is un
reachable.
•
The best approximate model (
g
): best descriptive model given the limited sample size. ±ind
ing the best
g
is (or can be) the goal of model selection.
•
A more parameterrich model has a higher potential than a less parameter rich model: less
discrepancy due to approximation
. However, a more parameterrich model tends to perform
farther below its potential than a less parameter rich model caused by the
discrepancy due to
estimation
[20].
•
Parsimonious trade o² between error (decreases with additional parameters) and variance
(increases with additional parameters).
•
To help us with the trade o²: apply a model selection criterion.
2
Model Selection Criteria in Phylogenetics
2.1
Likelihood
•
Changing the model changes the likelihood  (which is proportional to) the probability of
data, given the parameters and the model [5].
1
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Computational Evolutionary Biology
•
Maximized log Likelihood is biased upward as an estimator of the target model. The bias is
proportional to the number of parameters [4].
2.2
Likelihood Ratio Testing [7]
δ
=

2(ln
L
0

ln
L
1
)
•
Basic idea: Is the increase in likelihood signi±cant?
•
δ
is asymptotically
χ
2
n
distributed, with df
n
, the di²erence in number of free parameters
between models.
•
Only for nested models (model
L
0
must be a special case of
L
1
).
•
Mixed
χ
2
n
distributions when one parameter is in its limit (e.g., GTR vs. GTR+G) [19].
Applications:
Modeltest
[13],
MrModeltest2
[15]
2.3
AIC  Akaike Information Criterion [1, 4]
AIC
i
=

2 ln(
L
) + 2
p
•
L
: Max. log Likelihood for model
i
,
p
: number of parameters.
•
Estimates the expected KullbackLeibler (KL) distance: information lost when model
g
is
used to approximate
f
.
•
Min AIC is the best KL model in the set of competing models.
•
No accept or reject (not a strict test).
•
Applies to nested and nonnested models.
•
AIC
c
 takes sample size in to account.
•
Must be based on the maximum likelihood  problematic(?).
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 Spring '08
 staff
 Genetics, Conditional Probability, Evolution, Likelihood function, Bayesian statistics, Akaike information criterion, Computational Evolutionary Biology

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