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The AIC is defined as $AIC=-2 \log(L(\hat\theta))+2p$, where $\hat\theta$ is the maximum likelihood estimator and $p$ is the dimension of the parameter space. For the estimation of $\theta$, one usually neglects the constant factor of the density. This is, the factor that does not depend on the parameters, in order to simplify the likelihood. On the other hand, this factor is very important for the calculation of the AIC, given that when comparing non-nested models this factor is not common and then the order of the corresponding AICs might be different if it is not considered.

My question is, do we need to compute $\log(L(\hat\theta))$ including all the terms of the density when comparing non-nested models?

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  • $\begingroup$ I think I'm misunderstanding something. Where you say "For the estimation of $\theta$", did you mean "$L(\hat\theta)$"? $\endgroup$
    – David J. Harris
    Commented Jan 9, 2013 at 4:50
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    $\begingroup$ Since it's the difference in log-likelihood that matters, the terms that are in common are irrelevant, while any that different will matter. $\endgroup$
    – Glen_b
    Commented Jan 9, 2013 at 9:19

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Where the normalizing 'constant' differs across the models under consideration, those terms would need to be included.

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    $\begingroup$ Yes, that is what I think as well. Do you know of any reference on this? $\endgroup$
    – Kawabata
    Commented Sep 21, 2012 at 12:53
  • $\begingroup$ No references are needed, this is the definition of AIC which involves the exact loglikelihood (including those terms). $\endgroup$
    – Car Loz
    Commented Nov 15, 2023 at 0:17

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