Akaike Information Criterion (AIC)

This method deals with the modelling of a probability distribution of a random variable X. It seeks to rank variable candidate distributions by using a sample of data \left\{ \vect{x}_1,\vect{x}_2,\ldots,\vect{x}_N \right\}.

We denote by \cM_1,…, \cM_K the parametric models envisaged by user among the parametric models. We suppose here that the parameters of these models have been estimated previously by Maximum Likelihood the on the basis of the sample \left\{ \vect{x}_1,\vect{x}_2,\ldots,\vect{x}_n \right\}. We denote by L_i the maximized likelihood for the model \cM_i.

By definition of the likelihood, the higher L_i, the better the model describes the sample. However, using the likelihood as a criterion to rank the candidate probability distributions would involve a risk: one would almost always favor complex models involving many parameters. If such models provide indeed a large numbers of degrees-of-freedom that can be used to fit the sample, one has to keep in mind that complex models may be less robust that simpler models with less parameters. Actually, the limited available information (N data points) does not allow to estimate robustly too many parameters.

The Akaike Information Criterion (AIC) can be used to avoid this problem. The principle is to rank \cM_1,\dots,\cM_K according to the following quantity:

\begin{aligned} \textrm{AIC}_i = -2 \frac{\log(L_i)}{n} + \frac{2 p_i}{n} \end{aligned}

where p_i denotes the number of parameters being adjusted for the model \cM_i. The smaller \textrm{AIC}_i, the better the model. Note that the idea is to introduce a penalization term that increases with the numbers of parameters to be estimated. A complex model will then have a good score only if the gain in terms of likelihood is high enough to justify the number of parameters used.

In context of small data, there is a substantial risk that AIC select models that have too many parameters. In other words, the risk of overfitting is important. To tackle such issue, the AICc criterion was developed : it consists in evaluating the AIC with a correction term ( extra penalty) for small data. The formula is as follows :

\begin{aligned} \textrm{AICC}_i = AIC + \frac{(2 p_i)(p_i + 1)}{n - p_i - 1} \end{aligned}

One might notice that the extra term penalty vanishes for n \rightarrow \infty.

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