akaike_info_criterion¶

astropy.stats.akaike_info_criterion(log_likelihood, n_params, n_samples)[source]

Computes the Akaike Information Criterion (AIC).

Like the Bayesian Information Criterion, the AIC is a measure of relative fitting quality which is used for fitting evaluation and model selection. The decision is in favor of the model with the lowest AIC.

AIC is given as

$\mathrm{AIC} = 2(k - L)$

in which $$n$$ is the sample size, $$k$$ is the number of free parameters, and $$L$$ is the log likelihood function of the model evaluated at the maximum likelihood estimate (i. e., the parameters for which L is maximized).

In case that the sample size is not “large enough” a correction is applied, i.e.

$\mathrm{AIC} = 2(k - L) + \dfrac{2k(k+1)}{n - k - 1}$

Rule of thumb [1]:

$$\Delta\mathrm{AIC}_i = \mathrm{AIC}_i - \mathrm{AIC}_{min}$$

$$\Delta\mathrm{AIC}_i < 2$$: substantial support for model i

$$3 < \Delta\mathrm{AIC}_i < 7$$: considerably less support for model i

$$\Delta\mathrm{AIC}_i > 10$$: essentially none support for model i

in which $$\mathrm{AIC}_{min}$$ stands for the lower AIC among the models which are being compared.

For detailed explanations see [1]-[6].

Parameters
log_likelihoodfloat

Logarithm of the likelihood function of the model evaluated at the point of maxima (with respect to the parameter space).

n_paramsint

Number of free parameters of the model, i.e., dimension of the parameter space.

n_samplesint

Number of observations.

Returns
aicfloat

Akaike Information Criterion.

References

1(1,2)

Cavanaugh, J. E. Model Selection Lecture II: The Akaike Information Criterion. <http://machinelearning102.pbworks.com/w/file/fetch/47699383/ms_lec_2_ho.pdf>

2

Mazerolle, M. J. Making sense out of Akaike’s Information Criterion (AIC): its use and interpretation in model selection and inference from ecological data. <https://corpus.ulaval.ca/jspui/handle/20.500.11794/17461>

3

Wikipedia. Akaike Information Criterion. <https://en.wikipedia.org/wiki/Akaike_information_criterion>

4

Origin Lab. Comparing Two Fitting Functions. <https://www.originlab.com/doc/Origin-Help/PostFit-CompareFitFunc>

5

Liddle, A. R. Information Criteria for Astrophysical Model Selection. 2008. <https://arxiv.org/pdf/astro-ph/0701113v2.pdf>

6

Liddle, A. R. How many cosmological parameters? 2008. <https://arxiv.org/pdf/astro-ph/0401198v3.pdf>

Examples

The following example was originally presented in [2]. Basically, two models are being compared. One with six parameters (model 1) and another with five parameters (model 2). Despite of the fact that model 2 has a lower AIC, we could decide in favor of model 1 since the difference (in AIC) between them is only about 1.0.

>>> n_samples = 121
>>> lnL1 = -3.54
>>> n1_params = 6
>>> lnL2 = -4.17
>>> n2_params = 5
>>> aic1 = akaike_info_criterion(lnL1, n1_params, n_samples)
>>> aic2 = akaike_info_criterion(lnL2, n2_params, n_samples)
>>> aic1 - aic2
0.9551029748283746


Therefore, we can strongly support the model 1 with the advantage that it has more free parameters.