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  "body": "\n## Summary\n\nVariance reduction is the family of techniques that make a [[Monte Carlo method|HWP11G3HTN]] more accurate for the same computing effort.\nThe error of a plain Monte Carlo estimate with $n$ samples is about $\\sigma/\\sqrt{n}$, where $\\sigma^2$ is the [[variance|0477R9PYAE]] of one sample.\nThe rate $n^{-1/2}$ is fixed for plain random sampling.\nSo the gains come from replacing the sampled quantity by another with the same mean and a smaller variance.\nHalving $\\sigma$ saves a factor of four in the number of samples.\n\n## Explanations\n\n### L1 human\n\nA Monte Carlo estimate is noisy, and the noise shrinks slowly as more samples are taken.\nVariance reduction attacks the noise at its source: it changes what is sampled so that each sample is less scattered while the average stays the same.\nFor example, to estimate the average of $f(U) = U$ for uniform $U$ in $(0, 1)$, the pair average $\\frac12 (U + (1 - U))$ is always exactly $\\frac12$, so it has no scatter at all.\nReal problems do not give such perfect cancellation, but gains by factors of ten or a hundred are common.\n",
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