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  "body": "\n## Definition\n\nLet $M$ be a parametric statistical model with $k$ free real parameters, and let $x$ be data.\nLet $\\hat L$ be the maximum over the parameters of the [[likelihood function|05FSTXM92W]] of $M$ at $x$, the value at the [[maximum likelihood estimate|0FREPFKNS7]].\nThe *Akaike information criterion* of $M$ at $x$ is\n$$\\mathrm{AIC} = 2k - 2 \\ln \\hat L.$$\nAmong several candidate models for the same data, the AIC rule selects a model with the smallest AIC.\n\n## Remarks\n\nOnly differences of AIC between models fitted to the same data have a meaning.\nExample: a model with $k = 2$ and $\\ln \\hat L = -100$ has $\\mathrm{AIC} = 204$.\nA larger model with $k = 3$ and $\\ln \\hat L = -99.5$ has $\\mathrm{AIC} = 205$, so the AIC rule keeps the smaller model.\nNon-example: $-2 \\ln \\hat L$ alone, without the term $2k$, is not the AIC; it always favours the larger of two nested models.\nFor small samples a corrected version, AICc, adds a further penalty.\nThe [[Bayesian information criterion|GXP6H9VGRB]] uses the penalty $k \\ln n$ in place of $2k$.\n",
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