Compare GLM AICs with different likelihoods? If I have a generalized linear model (GLM) with a particular likelihood, and I have another GLM of the same data (say nested within the first model), I can compare the model performance using Akaike information criterion (AIC).
Let's say that I fit a Poisson GLM.
However, I wonder if I might get a better fit by using a negative binomial distribution.
Assuming the same features, what, if any, meaning is there to the difference in AIC values between the Poisson model and the negative binomial model? That is:
$$
\log(\mathbb E[Y\vert X]) = \alpha_0 +\alpha_1X \implies AIC_1\text{, based on Poisson likelihood}\\
\log(\mathbb E[Y\vert X]) = \beta_0 +\beta_1X \implies AIC_2\text{, based on negative binomial likelihood}
$$
Is there any meaning to comparing $AIC_1$ and $AIC_2?$ The answer here seems to indicate that it is not meaningful to compare $AIC_1$ and $AIC_2$, but the comments express some dissent. Kjetil's answer here seems to indicate that such a comparison would not be meaningful for a Poisson vs a Gaussian (for example) likelihood, due to the discrete vs continuous dominating measures, but for Poisson vs negative binomial, both dominating measures are discrete.
(A similar setup would be if it is meaningful to compare a linear regression fitted by minimizing square loss (maximum likelihood estimation for a Gaussian likelihood) and a linear regression fitted by minimizing absolute loss (maximum likelihood estimation for a Laplace likelihood). That seems like comparing MSE and MAE (or RMSE and MAE), which seems like an unfair comparison. At the same time, both dominating measures would be continuous.)

If we dissect AIC formula by Akaike (1974),
$$
AIC=-2\ln(L)+2k,
$$
where $k$ is the number of parameters, the two differences between $AIC_1$ and $AIC_2$ are the value of $k$ (negative binomial likelihood will have an extra parameter) and the $\ln(L)$ values.
Both are discrete regression models and they are often used to compare two or more models. However, one may argue if the difference in $AIC_1$ and $AIC_2$ are significant enough to conclude one model is better than the other. In this case, one may employ a likelihood ratio test and decide which model is the best.
See Likelihood Ratio Test for Poisson vs NB GLM
