# How to use AIC to estimate relative support for models factoring in sample size?

I am having a problem understanding how to estimate support for models when the number of samples is different. The scenario is as follows:

Suppose the log-likelihood of models $M_1$ and $M_2$ of some data $\mathbf{x}$ are $\ell_1$ and $\ell_2$ respectively, and $\ell_1 > \ell_2$, and that both models have the same number of estimated parameters. Then, $M_2$ is $\exp(\ell_2 - \ell_1)$ times as probable as the first model to minimise information loss from Wikipedia (and discussed here).

Now consider that $\mathbf{z}$ is the dataset consisting of replicating the observations in $\mathbf{x}$ twice. Then, models $M_1$ and $M_2$ have log-likelihoods equal to $2\ell_1$ and $2\ell_2$ respectively, and $M_2$ is $\exp(2(\ell_2 - \ell_1))$ times as probable as the first model to minimise information loss.

This gives relative probability different to the one first calculated. How can this be so?

• Replicating the samples is just a convenient way of showing the mathematical calculation. we can assume we know the true generation process, then draw twice as many samples.
– Alex
Aug 8 '17 at 11:51

• Thanks, I corrected the wording in my question as you are right. However, there is no factor of 0.5 as the AIC has the term $2\ell$ in it.