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  "body": "\n## Summary\n\nThe Fisher information $I(\\theta)$ measures how much an observation tells about an unknown parameter $\\theta$ of a statistical model.\nIt is the variance of the score, the derivative in $\\theta$ of the log-likelihood.\nIt bounds the precision of unbiased estimators through the [[Cramér–Rao bound|1JFKTEQVRX]], and its inverse is the asymptotic variance of the [[maximum likelihood estimator|0FREPFKNS7]].\n\n## Explanations\n\n### L1 human\n\nIf a small change of the parameter changes the distribution of the data a lot, then the data pin the parameter down well, and the information is large.\nIf the distribution hardly moves, the information is small.\nPicture the log-likelihood as a hill: high information means a sharp peak, low information a flat top.\nFor one observation of a normal variable with mean $\\theta$ and variance $\\sigma^2$, the information is $1/\\sigma^2$, so less noise means more information.\nWith $n$ independent observations the information adds up to $n/\\sigma^2$.\n",
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