### Akaike information criterion

**(AIC)**
(particular formula for comparison of statistical models)

The **Akaike information criterion** (**AIC**) is a calculated
value (a "score") associated with a statistical model
and sample of data, which is intended to be used
to compare the utility of competing statistical models
in light of the data.
It is used by calculating the AIC of two models applied to the
same sample data and comparing the scores of the two.
Formula:

AIC = 2k - 2 ln(maxL)

- k - number of parameters in the model.
- maxL - maximum likelihood of the model's distribution.

Lower score is better. An example of the type of model that
might be evaluated is an initial mass function (IMF).

Given some collected data, a statistical model (such as probability density function (PDF) or
probability mass function (PMF)) can be developed aiming
to match the "real" process that generated the data. Types of such
models are unlimited: they can be polynomials of any degree, and
can incorporate other functions such as powers, logs, roots,
trigonometric functions, etc. Given functions of the same form
except for constants, a criteria such as least squares is useful
to choose between them, but given the other choices, something else
is needed. A polynomial of sufficiently high degree can be constructed
to exactly match the distribution of any sample data, and short of
that, models with more parameters can come closer to matching the
distribution. Criteria is needed to decide whether the model is
**overfitted**, i.e., whether the model is so specific to the sample
upon which it was based that it is unlikely to fit another sample
from the same source. The aim of the AIC is to comparatively
evaluate the models for such overfitting.

AIC is reasonable for certain kinds of models, and is tailored to
large samples (many data points). A modified AIC (**AIC corrected**
or **corrected AIC**, abbreviated **AICc**) essentially includes a
second-order term which improves the score and is most useful given
small samples.

**Deviance information criterion** (**DIC**) is a generalized version
of AIC for similarly comparing **hierarchical statistical models**
(**multilevel statistical models**), for modeling processes that
have more than one source of variation. I think an example of
this might be an IMF that also depends upon cosmological redshift.

(*statistics*)
**Further reading:**

http://en.wikipedia.org/wiki/Akaike_information_criterion

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