# 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 -

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).
• I don't think it makes much sense. What criterion do you use to estimate the parameters $m$ and $c$? Least squares? Jul 31, 2018 at 23:10
• Given the way in which it is defined, $s$ seems to represent the average distance between the model predictions $\hat y$ and the actual value of $y$, so it isn't really measurement noise, nor some noise in the generative process that generated the data. Jul 31, 2018 at 23:17