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  "body": "\n## Summary\n\nA rate function is a lower semicontinuous function $I$ from a topological space to $[0, \\infty]$.\nIn [[large deviations theory|Q702WCDQQ3]], it gives the exponential rate at which the probability of a rare event decays: roughly, $P(Z_n \\approx x) \\approx e^{-n I(x)}$.\nIt is called good when its sublevel sets are compact.\nExamples are the Legendre transforms in [[Cramér's theorem|RZZK51CXMN]] and the [[Kullback–Leibler divergence|PQBW43X5RE]] in [[Sanov's theorem|QWEF2WAXJM]].\n\n## Explanations\n\n### L1 human\n\nA rate function measures how costly it is for a random system to end up at a given state.\nThe typical state costs nothing, and states far from typical cost more.\nThe probability of a state falls like $e^{-n \\cdot \\text{cost}}$, where $n$ is the size of the system.\nFor the average of $n$ standard normal variables, the cost of the value $x$ is $x^2/2$.\n",
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