Carl Boettiger wrote:
Others on the list can weigh in with more authority, but perhaps this will
get the discussion started.
Yes, it's important to know whether the parameters are nested, and
the issue of being at the end of a parameter range is serious.
Recall that AIC values are a fequentist statistic: and they obey the very
same same distribution as the likelihood ratio, (recall it is a difference
log likelihoods, just shifted by the difference in the number of parameters
(e.g. -2 [ log L1 - log L0 - (k1 - k0)]). Recall that the maximum
likelihood estimate (MLE) is a biased estimate of the likelihood of your
data and that AIC penalty is simply creating an asymptotically unbiased**
estimator of the true model likelihood, which is a frequentist concept to
begin with. Why we report confidence intervals/p-values in the case of one
of these statistics but not the other is not obvious to me either.
I will confess my relative ignorance of AIC issues (my phylogeny book
has a simple, elegant, and clear explanation -- which I wrote in a hurry
while excited that I finally understood this, and which turns out to
make no sense whatsoever and should be firmly ignored by all).
But I do know this: If we have the likelihood ratio R = L(p')/L(p) where
p' is the ML parameter values and p is the true parameter values,
and where p is in the interior of the set of possible parameters,
then RA Fisher showed about 1922 that asymptotically with large
amounts of data:
2 log(R) is distributed as chi-square with D degrees of freedom,
where D is the difference of the number of parameters being
estimated in p' and the number of parameters being estimated in
p. Now we know that the expectation of that chi-squared variable
is D. So to correct the bias in R we should subtract D. That sounds
like what Carl is explaining too.
It sounds like a very simple and clear explanation of the AIC. Unfortunately
that subtraction is *not* what AIC does. It subtracts 2D. The reason
it does so is unclear to me. It involves some kind of prior on models,
I think. As far as I am concerned it is like the peace of god, in that it
passeth human understanding.
Maybe the experts here can give me a simple explanation. Otherwise
maybe we should honor Fisher (not me) and only subtract D, and call the
result the FIC, But that works only for nested hypotheses, and the main
point of the AIC is to deal with non-nested hypotheses. To make matters
worse, in my field the AIC has the reputation of too easily favoring the
most complex hypothesis, so maybe we should be subtracting more than
2D, not less.
Clueless in Seattle.
Joe
Joe Felsenstein j...@gs.washington.edu
Department of Genome Sciences and Department of Biology,
University of Washington, Box 355065, Seattle, WA 98195-5065 USA
[[alternative HTML version deleted]]
___
R-sig-phylo mailing list
R-sig-phylo@r-project.org
https://stat.ethz.ch/mailman/listinfo/r-sig-phylo
