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  "body": "\n## Statement\n\nLet $\\mathcal{X}$ be a compact metric space, the design region.\nLet $f \\colon \\mathcal{X} \\to \\R^p$ be continuous, and consider the [[linear regression|41FVC4P5JB]] model $E[y(x)] = \\theta^T f(x)$ with $\\theta \\in \\R^p$ and uncorrelated errors of equal variance.\nA design is a [[probability measure|059TJN0NZ1]] $\\xi$ on the Borel sets of $\\mathcal{X}$.\nIts information matrix is $M(\\xi) = \\int_{\\mathcal{X}} f(x) f(x)^T \\, d\\xi(x)$.\nWhen $M(\\xi)$ is nonsingular, its standardized variance function is $d(x, \\xi) = f(x)^T M(\\xi)^{-1} f(x)$.\nAssume that some design has a nonsingular information matrix.\nFor a design $\\xi^*$ with $M(\\xi^*)$ nonsingular, the following three conditions are equivalent:\n\n1. $\\xi^*$ is D-optimal: it maximizes $\\det M(\\xi)$ over all designs $\\xi$.\n2. $\\xi^*$ is G-optimal: it minimizes $\\max_{x \\in \\mathcal{X}} d(x, \\xi)$ over all designs $\\xi$ with $M(\\xi)$ nonsingular.\n3. $\\max_{x \\in \\mathcal{X}} d(x, \\xi^*) = p$.\n\nMoreover, all D-optimal designs have the same information matrix, and $d(x, \\xi^*) = p$ at every support point $x$ of a D-optimal design $\\xi^*$.\n\n## Remarks\n\nFor every design with nonsingular information matrix, $\\int d(x, \\xi) \\, d\\xi(x) = \\tr(M(\\xi)^{-1} M(\\xi)) = p$.\nSo $\\max_x d(x, \\xi) \\ge p$ always holds, and this explains the number $p$ in condition 3.\nThe function $\\xi \\mapsto \\log \\det M(\\xi)$ is concave on the convex set of designs, and its directional derivative at $\\xi$ towards the point mass $\\delta_x$ equals $d(x, \\xi) - p$.\nSo condition 3 says that no direction increases $\\log \\det M$, which for a concave function means a maximum.\nExample: for the line $f(x) = (1, x)^T$ on $\\mathcal{X} = [-1, 1]$, the design $\\xi^* = \\frac12 \\delta_{-1} + \\frac12 \\delta_{1}$ has $M(\\xi^*) = I_2$.\nThen $d(x, \\xi^*) = 1 + x^2 \\le 2 = p$, so $\\xi^*$ is D-optimal and G-optimal.\nThe theorem was proved by Jack Kiefer and Jacob Wolfowitz in 1960.\n\n## Explanations\n\n### L1 human\n\nAn experimenter chooses where to take measurements in order to fit a regression model.\nOne goal is to make the joint uncertainty about the parameters small, measured by the [[determinant|00KDZRW4C0]] of the [[information matrix|G1Z3WSTE8Q]].\nAnother goal is to make the worst prediction variance over the whole region small.\nThe theorem says that these two goals give the same best designs, and it gives a simple test for being best.\n\n### L2 human\n\nThe equivalence theorem turns a hard optimization over probability measures into a checkable condition: compute $d(x, \\xi)$ and see whether it stays below $p$.\nIt is the basis of the Fedorov–Wynn type algorithms, which add weight at the point where $d(x, \\xi)$ is largest.\nSimilar equivalence theorems hold for other concave criteria of [[optimal experimental design|PM5PQX6TVP]], with the variance function replaced by the corresponding directional derivative.\n",
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