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 [email protected] 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 [email protected] https://stat.ethz.ch/mailman/listinfo/r-sig-phylo
