Why is AIC not reported with a confidence interval? In parameter estimation, it's common to report a 95% CI around each parameter. Why don't I see AIC (or deltaAIC) with a CI?
If I bootstrap the fitting of two potential models, and get a deltaAIC for each iteration, would reporting a 95% CI be meaningful?
Edit: I don't mean to imply that AIC is the same as parameter estimation. I'm asking why we treat goodness-of-fit estimates (AIC, BIC, etc.) differently from estimates that are reported with a CI.
 A: The AIC is not an estimator of a true parameter. It is a data-dependent measurement of the model fit. The model fit is what it is, there is no model fit that is any "truer" than the one you have, because it's the one you have that is measured. But without any true parameter for which the AIC would be an estimator, one cannot have a confidence interval (CI).
I'm by the way not disputing the answer by Richard Hardy. The AIC, as some other quantities such as $R^2$, can be interpreted as estimating something "true but unobservable", in which case one can argue that a CI makes sense. Personally I find the interpretation as measuring fit quality more intuitive and direct, for which one wouldn't have a CI for the reasons above, but I'm not saying that there is no way for it to be well defined and of some use.
Edit: As a response to the addition in the question: "I don't mean to imply that AIC is the same as parameter estimation. I'm asking why we treat goodness-of-fit estimates (AIC, BIC, etc.) differently from estimates that are reported with a CI." - The definition of a CI relies on a parameter being estimated. It says that given the true parameter value the CI catches this value with probability $(1-\alpha)$. As long as you're not interested in that true parameter value, a CI is meaningless.
A: AIC estimates $-2n \ \times$ the expected likelihood on a new, unseen data point from the data generating process (DGP) that generated your sample.* Even though the target (the estimand) is not a parameter, it is a meaningful quantity. E.g. it may be interpreted as the expected loss of a point prediction. It is quite natural to wish for a confidence interval around the point estimate (the AIC). This way we can tell not only what the expected loss is but also how uncertain it is. In summary, while I do not have a ready answer for how to obtain the confidence interval and under what conditions your idea of bootstrapping may work, I clearly do see a point in pursuing it.
*See How can we select the best GARCH model by carrying out likelihood ratio test?, Can results for model selection with AIC be interpretable at the population level?, Using AIC/BIC within cross-validation for likelihood based loss functions among other threeads where this idea is employed.
