Two GLMs of the same distribution are fit to the same set of data, using Maximum Likelihood Estimation (MLE) to fit the regression parameters $\beta$. The scale/dispersion parameter for each model is then separately estimated (using the same estimation, but not necessarily MLE).
Is it appropriate to calculate the AIC for each model, using the different scale parameters for each model, and use the difference to choose between them?
My motivation for this question is that for nested models $M_1 \subset M_2$, the differences in estimated scale parameters can make the calculated log likelihood of $M_1$ larger than $M_2$. My intuition for AIC is that the log likelihood measures how well we fit the data, and this is penalized by model complexity measured as 2 times the number of parameters. But that intuition appears to be wrong in this case, since the log likelihood of the more complicated model decreased all by itself, owing to the change in scale parameter.
As an aside (and potentially an answer), I assume the situation I mention could not happen if the scale parameter were also fit with MLE. Is it that such comparisons are valid only when the scale parameter is fit with MLE?
