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.