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  "body": "\n## Statement\n\nLet $f \\colon (0, 1) \\to \\R$ be monotone (non-decreasing or non-increasing) with $\\int_0^1 f(u)^2\\, du \u003c \\infty$, and let $U$ have the [[uniform distribution|0AREFQ9Z65]] on $(0, 1)$.\nLet $Z = \\frac12 \\bigl( f(U) + f(1 - U) \\bigr)$.\nThen:\n\n1. $\\mathbb E Z = \\mathbb E f(U) = \\int_0^1 f(u)\\, du$;\n2. $\\operatorname{Cov}\\bigl(f(U), f(1 - U)\\bigr) \\le 0$;\n3. $\\operatorname{Var} Z \\le \\frac12 \\operatorname{Var} f(U)$.\n\nBoth $n$ copies of $Z$ and $2n$ copies of $f(U)$ need $2n$ evaluations of $f$.\nThe average of $n$ independent copies of $Z$ is at least as accurate as that of $2n$ independent copies of $f(U)$.\n\n## Remarks\n\nClause 1 holds because $1 - U$ is uniform on $(0, 1)$ too.\nFor clause 2, if $f$ is non-decreasing then $u \\mapsto f(1 - u)$ is non-increasing, and for a non-decreasing $a$ and a non-increasing $b$ one has $\\mathbb E[a(U) b(U)] \\le \\mathbb E a(U)\\, \\mathbb E b(U)$, the integral form of [[Chebyshev's sum inequality|S8X2N1DW8P]].\nThis in turn follows from $\\mathbb E\\bigl[(a(U) - a(V))(b(U) - b(V))\\bigr] \\le 0$ for an independent copy $V$ of $U$, since the two factors have opposite signs.\nClause 3: $\\operatorname{Var} Z = \\frac14 \\bigl( 2 \\operatorname{Var} f(U) + 2 \\operatorname{Cov}(f(U), f(1-U)) \\bigr) \\le \\frac12 \\operatorname{Var} f(U)$, using $\\operatorname{Var} f(1 - U) = \\operatorname{Var} f(U)$.\nExample: for $f(u) = e^u$, $\\operatorname{Var} f(U) \\approx 0.2420$ while $\\operatorname{Var} Z \\approx 0.0039$, a gain of a factor of about $30$ over $2$ independent samples.\nMonotonicity is needed: for $f(u) = (u - \\frac12)^2$ one has $f(1 - u) = f(u)$, so $Z = f(U)$ and $\\operatorname{Var} Z = \\operatorname{Var} f(U)$, twice the variance of the average of two independent samples.\n\n## Explanations\n\n### L1 human\n\nThe idea is to sample in pairs that are opposite to each other: $U$ and $1 - U$.\nIf one value of the function is too high, its partner tends to be too low, and the errors partly cancel in the average.\nFor a function that always increases or always decreases, the method is guaranteed not to hurt and often helps a lot.\n\n### L2 human\n\nAntithetic variates is one of the simplest [[variance reduction|A01GEGWKVB]] techniques, with no extra modelling and almost no extra cost.\nThe same idea applies to normal inputs with the pair $Z$ and $-Z$, and to simulations driven by many uniforms, replacing each $U_j$ by $1 - U_j$.\nIn the multivariate case the guarantee holds when the output is a monotone function of each input separately, but for general outputs the method can increase the variance.\nCombined with inverse transform sampling, which is monotone, it yields negatively correlated pairs from any distribution.\n",
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