Akaike information criterion
The Akaike information criterion is a measure of the relative quality of a statistical model, for a given set of data. As such, AIC provides a means for model selection. AIC is founded on information entropy: it offers a relative estimate of the information lost when a given model is used to represent the process that actually generates the data.
AIC deals with the trade-off between the complexity of the model and the goodness of fit of the model.
AIC does not provide a test of a model in the sense of testing a null hypothesis; i.e. AIC can tell nothing about how well a model fits the data in an absolute sense. If all the candidate models fit poorly, AIC will not give any warning of that.
In the general case, the AIC is
Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value. Hence AIC not only rewards goodness of fit, but also includes a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting (increasing the number of free parameters in the model improves the goodness of the fit, regardless of the number of free parameters in the data-generating process).
AIC is founded in information theory. Suppose that the data is generated by some unknown process f. We consider two candidate models to represent f: g1 and g2. If we knew f, then we could find the information lost from using g1 to represent f by calculating the Kullback–Leibler divergence, DKL(f ‖ g1); similarly, the information lost from using g2 to represent f would be found by calculating DKL(f ‖ g2). We would then choose the candidate model that minimized the information loss.
We cannot choose with certainty, because we do not know f. Akaike (1974) showed, however, that we can estimate, via AIC, how much more (or less) information is lost by g1 than by g2. It is remarkable that such a simple formula for AIC results. The estimate, though, is only valid asymptotically; if the number of data points is small, then some correction is often necessary (see AICc, below).
How to apply AIC in practice 
To apply AIC in practice, we start with a set of candidate models, and then find the models' corresponding AIC values. There will almost always be information lost due to using one of the candidate models to represent the "true" model (i.e. the process that generates the data). We wish to select, from among R candidate models, the model that minimizes the information loss. We cannot choose with certainty, but we can minimize the estimated information loss.
Denote the AIC values of the candidate models by AIC1, AIC2, AIC3, …, AICR. Let AICmin be the minimum of those values. Then exp((AICmin−AICi)/2) can be interpreted as the relative probability that the ith model minimizes the (estimated) information loss.
As an example, suppose that there were three models in the candidate set, with AIC values 100, 102, and 110. Then the second model is exp((100−102)/2) = 0.368 times as probable as the first model to minimize the information loss, and the third model is exp((100−110)/2) = 0.007 times as probable as the first model to minimize the information loss. In this case, we would omit the third model from further consideration. We could take a weighted average of the first two models, with weights 1 and 0.368, respectively, and then do statistical inference based on the weighted multimodel; alternatively, we could gather more data to distinguish between the first two models.
The quantity exp((AICmin−AICi)/2) is the relative likelihood of model i.
If all the models in the candidate set have the same number of parameters, then using AIC might at first appear to be very similar to using the likelihood-ratio test. There are, however, important distinctions. In particular, the likelihood-ratio test is valid only for nested models whereas AIC (and AICc) has no such restriction.
AICc is AIC with a correction for finite sample sizes:
where n denotes the sample size. Thus, AICc is AIC with a greater penalty for extra parameters.
Burnham & Anderson (2002) strongly recommend using AICc, rather than AIC, if n is small or k is large. Since AICc converges to AIC as n gets large, AICc generally should be employed regardless. Using AIC, instead of AICc, when n is not many times larger than k2, increases the probability of selecting models that have too many parameters, i.e. of overfitting. The probability of AIC overfitting can be substantial, in some cases.
Brockwell & Davis (1991, p. 273) advise using AICc as the primary criterion in selecting the orders of an ARMA model for time series. McQuarrie & Tsai (1998) ground their high opinion of AICc on extensive simulation work with regression and time series.
AICc was first proposed by Hurvich & Tsai (1989). Different derivations of it are given by Brockwell & Davis (1991), Burnham & Anderson, and Cavanaugh (1997). All the derivations assume a univariate linear model with normally distributed errors (conditional upon regressors); if that assumption does not hold, then the formula for AICc will usually change. Further discussion of this, with examples of other assumptions, is given by Burnham & Anderson (2002, ch. 7). In particular, bootstrap estimation is usually feasible.
Note that when all the models in the candidate set have the same k, then AICc and AIC will give identical (relative) valuations. In that situation, then, AIC can always be used.
Relevance to chi-squared fitting 
Often, one wishes to select amongst competing models where the likelihood functions assume that the underlying errors are normally distributed (with mean zero) and independent. This assumption leads to model fitting.
For fitting, the likelihood is given by
where C is a constant independent of the model used, and dependent only on the use of particular data points. i.e. it does not change if the data do not change.
The AIC is therefore given by . As only differences in AIC are meaningful, the constant C can be ignored, allowing us to take for model comparisons.
Another convenient form arises if the σi are assumed to be identical and the residual sum of squares (RSS) is available. Then we get AIC = n ln(RSS/n) + 2k + C, where again C can be ignored in model comparisons.
The original derivation of AIC relied upon some strong assumptions. Takeuchi (1976) showed that the assumptions could be made much weaker. This work, however, was in Japanese, and was not widely known outside Japan for many years.
AICc was originally proposed for linear regression (only) by Sugiura (1978). That instigated the work of Hurvich & Tsai (1989), and several further papers by the same authors, which extended the situations in which AICc could be applied. The work of Hurvich & Tsai contributed to the decision to publish a second edition of the volume by Brockwell & Davis (1991), which is the standard reference for linear time series; the new edition states, "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc".
The volume by Burnham & Anderson (2002) was the first attempt to set out the information-theoretic approach in a general context. It includes an English exposition of the results of Takeuchi. The volume led to far greater use of the information-theoretic approach, and now has over 17000 citations on Google Scholar.
Akaike originally called his approach an “entropy maximization principle”. Burnham & Anderson (2002, ch. 2) discuss and expand on this, and trace the approach back to the work of Ludwig Boltzmann on thermodynamics. Briefly, minimizing AIC in a statistical model is essentially equivalent to maximizing entropy in a thermodynamic system. In other words, the information-theoretic approach in statistics is essentially applying the Second Law of Thermodynamics.
Bayesian information criterion 
The AIC penalizes the number of parameters less strongly than does the Bayesian information criterion (BIC), which was independently developed by Akaike and by Schwarz in 1978, using Bayesian formalism. Akaike's version of BIC was originally denoted ABIC (for "a Bayesian Information Criterion") or referred to as Akaike's Bayesian Information Criterion.
A comparison of AIC/AICc and BIC is given by Burnham & Anderson (2002, § 6.4). The authors argue that AIC/AICc has theoretical advantages over BIC. Firstly, because AIC/AICc is derived from principles of information. Secondly, because the (Bayesian) derivation of BIC has a prior of 1/R (where R is the number of candidate models), which is "not sensible", since the prior should be a decreasing function of k. The authors also show that AIC and AICc can be derived in the same Bayesian framework as BIC, just by using a different prior. Additionally, they present a few simulation studies that suggest AICc tends to have practical/performance advantages over BIC. See also Burnham & Anderson (2004).
Further comparison of AIC and BIC, in the context of regression, is given by Yang (2005). In particular, AIC is asymptotically optimal in selecting the model with the least mean squared error, under the assumption that the exact "true" model is not in the candidate set (as is virtually always the case in practice); BIC is not asymptotically optimal under the assumption. Yang further shows that the rate at which AIC converges to the optimum is, in a certain sense, the best possible.
See also 
- Deviance information criterion
- Focused information criterion
- Hannan–Quinn information criterion
- Occam's Razor
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