use of akaike information criterion with nonnested models Would anybody be able to explain to me why the likelihood function $L$ can not be used to compare nonnested models whereas the AIC, which is a simple function of $L$ ($-2\log L+2k$ where $k$ is the number of parameters), can?
In particular I am wondering how two models such as gamma and lognormal, with completely different likelihood functions can then be compared because of a correction of the number of parameters. I believe that these likelihood functions are on different scales because of different normalising constants and are therefore quite different.
It is my understanding (I may be quite wrong) that the AIC represents how close a model is to minimising the "distance" (entropy) to the true model. The likelihood function represents a different distance but a distance all the same. Why the sudden jump from nested to nonnested?
Thanks in advance!
 A: It seems that the issue of using AIC for non-nested models is not fully resolved. From what I can understand there are two schools of thought which I shall try to summarise, but as a non-expert:
1) Burnham & Anderson (Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.), Springer-Verlag, ISBN 0-387-95364-7) suggest that AIC can be used for non-nested models. In their derivation of the AIC it looks like they do not make too many assumptions on the relationship between the "true" distribution and the model distribution. Therefore they don't mind to compare quite (within reason) different models.
2) Ripley (Ripley, B. D. 2004. “Selecting amongst Large Classes of Models.” In Methods and Models in Statistics, edited by N. Adams, M. Crowder, D. J Hand, and D. Stephens, 155–70. London, England: Imperial College Press) argues that AIC can not be used for non-nested models. He suggests that the nested assumption is necessary to keep the variance of the estimator of the AIC low. If it is not the case then the estimate is not useable.
The authors have had a back-and-forth over the topic which is played out here. 
In my research I have found the following posts very useful:
https://stats.stackexchange.com/questions/116935/comparing-non-nested-models-with-aic
https://stats.stackexchange.com/questions/37666/aic-for-non-nested-models-normalizing-constant
https://stats.stackexchange.com/questions/20441/non-nested-model-selection
https://stat.ethz.ch/pipermail/r-help/2006-February/088745.html
Now, it may be that someone who can understand both the Ripley and the Burnham & Anderson derivations better than me will know the answer to my original question, but I don't believe the answer is very clear to wide community yet. I am however, still interested in why the issue is far more clear for the likelihood function; unfortunately I still do not understand the "jump" from nested to non-nested.
I will therefore use AIC to compare non-nested models but with caution and perhaps move on to different Monte Carlo methods suggested by Ripley at some point.
A: In theory, either can be used, even when the models are different.  The idea is that you use the Kullback-Liebler divergence to choose between the models.  You can estimate this by taking the sample log likelihood, and divide by the sample size.  It is known that this can be biased in small samples with a bias proportional to the number of parameters, so the AIC is an attempt to adjust for this bias.  In large samples, the correction (when divided by the sample size) becomes negligible. 
Note that if you want to any hypothesis testing, the asymptotic statistics for non-nested models is different.  The asymptotics were worked out by Vuong.
