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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 of $n$ observations.\nLet $\\hat L$ be the maximum of the [[likelihood function|05FSTXM92W]] of $M$ at $x$, the value at the [[maximum likelihood estimate|0FREPFKNS7]].\nThe *Bayesian information criterion* of $M$ at $x$ is\n$$\\mathrm{BIC} = k \\ln n - 2 \\ln \\hat L.$$\nAmong several candidate models for the same data, the BIC rule selects a model with the smallest BIC.\n\n## Remarks\n\nUnder regularity conditions, $-\\mathrm{BIC}/2$ approximates the logarithm of the marginal likelihood of $M$, up to a term that stays bounded as $n \\to \\infty$.\nSo differences of BIC approximate $-2$ times the logarithm of a Bayes factor.\nExample: with $n = 100$, a model with $k = 2$ and $\\ln \\hat L = -100$ has $\\mathrm{BIC} = 2 \\ln 100 + 200 \\approx 209.2$.\nA model with $k = 3$ and $\\ln \\hat L = -98$ has $\\mathrm{BIC} \\approx 209.8$, so the BIC rule keeps the smaller model, while the AIC values $204$ and $202$ favour the larger one.\nNon-example: $2k - 2 \\ln \\hat L$ is the [[Akaike information criterion|CMAX3KF9H0]], not the BIC; the two agree only when $\\ln n = 2$.\n",
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