Q: Number of AIC parameters (again) I know very similar questions get asked a lot - but I would like to make sure that I am not missing something here:
Suppose we have the simple model:
y = mx + c + s*e
where e ~ N(0,1) and m, c and s are model parameters.
In the case where we are holding 's' fixed the number of parameters used to calculate the AIC is 2 (m and c).
The reason for the question is that I have seen it written that the 'noise term' should be counted - but I presume that is really referring to the 's' parameter in this example, and so should not be included if it is being held fixed. The 'e' RV does not itself contribute anything.
Again, apologies for the similarity to other questions - just want to confirm this specific point.
Thanks - 
 A: In your example $s$ seems to be the standard deviation of the residuals. In that case it should be counted as a parameter and it doesn't make much sense to held it fixed. Let say you compare (on the same dataset) your model with a more complex model, e.g. $y = mx^2 + mx + c + se$, and to an even simpler model $y = c + se$. The predictions of the more complex model (that is $\hat y = mx^2 + mx + c$) will be on average closer to the actual values of $y$, while the predictions of the simpler model will be on average more off. Thus, the standard deviation of the residuals will be smaller for the more complex model, and larger for the simpler one. If you held it fixed then for both models the set of parameters is not the one that maximizes the likelihood of the data and therefore you cannot use the AIC (which is calculated from the maximum value of the likelihood function and the number of free parameters). 
Note also that the AIC is only a measure of the relative quality of statistical models (its absolute value doesn't mean much). In this example counting it or not in the calculation of the AIC amounts to adding  a constant, 2, to the AIC of all three models. Therefore the results of the model comparisons, e.g. expressed in terms of AIC differences, would be exactly the same
