9

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?

Community
  • 1
Zachary Blumenfeld
  • 3,826
  • 1
  • 14
  • 21
  • 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? – David G. Stork Jan 25 '15 at 08:21
  • 2
    Likelihoods *do not* need to be $\leq 1$, since densities can exceed 1. Indeed, [see here](http://en.wikipedia.org/wiki/Likelihood_function#Likelihoods_for_continuous_distributions): "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). – Glen_b Jan 25 '15 at 15:50

1 Answers1

11

$L$ 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.

Scortchi - Reinstate Monica
  • 27,560
  • 8
  • 81
  • 248
Richard Hardy
  • 54,375
  • 10
  • 95
  • 219