I'm not that familiar with this literature, so please forgive me if this is an obvious question.
Since AIC and BIC depend on maximizing the likelihood, it seems that they can only be used to make relative comparisons between a set of models attempting to fit a given data-set. According to my understanding, it wouldn't make sense to calculate the AIC for Model A on data-set 1, calculate the AIC for Model B on data-set 2, and then compare the two AIC values and judge that (for example) Model A fits data-set 1 better than Model B fits data-set 2. Or perhaps I am mistaken and that is a reasonable thing to do. Please let me know.
My question is this: does there exist a model fit statistic that can be used for absolute instead of just relative comparisons? For linear models, something like $R^2$ would work; it has a defined range and discipline specific ideas about what is a "good" value. I'm looking for something more general and thought that I could start by pinging the experts here. I'm sure that someone has thought of this kind of thing before, but I don't quite know the right terms to make a productive search on Google Scholar.
Any help would be appreciated.