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My question is related to the thread Negative values for AIC in General Mixed Model. I often get negative AIC values from the software I use. I notice it most when I'm doing time series. But here is what I don't get. When defining the AIC like

$$AIC = 2k-2\ln(L)$$

$L$, the likelihood, is a joint probability and to my understanding must be bound between 0 and 1. Mathematically this implies the $AIC$ must be positive. So I don't know what my software is giving me for the value labeled $AIC$. Any thoughts?

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  • $\begingroup$ What software do you use? Could you give us a specific example that yields a negative AIC so we can check using our software and analyses? $\endgroup$
    – David G. Stork
    Commented Jan 25, 2015 at 8:21
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    $\begingroup$ Likelihoods do not need to be $\leq 1$, since densities can exceed 1. Indeed, see here: "likelihood is only defined up to a multiplicative constant"; (only positive ones could make sense, though). Log-likelihoods only make sense when compared with other log likelihoods (the arbitrary shift must be the same for both, naturally). $\endgroup$
    – Glen_b
    Commented Jan 25, 2015 at 15:50

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$L$ is not a joint probability (joint cumulative probability density) but joint probability density. Since density only needs to be non-negative and is not bounded from above, $\operatorname{ln}(L)$ can be both positive and negative. Hence, $AIC$ can also be both positive and negative.

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    $\begingroup$ (+1) NB For a discrete response the likelihood will be a joint probability, provided you don't drop any constant factor necessary to keep it so. $\endgroup$
    – Scortchi ♦
    Commented Jan 25, 2015 at 15:35
  • $\begingroup$ Thanks for the help...Should have known this one, but doing so much probit/logit lately I forgot the greater realm of MLE. $\endgroup$
    – Zachary Blumenfeld
    Commented Jan 28, 2015 at 3:35

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