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  "body": "\n## Definition\n\nLet $(M, d)$ be a separable complete metric space, that is, a [[Polish space|584GXXY73M]], and let $p \\in [1, \\infty)$.\nLet $\\mathcal{P}_p(M)$ be the set of Borel [[probability measures|059TJN0NZ1]] $\\mu$ on $M$ with $\\int_M d(x, x_0)^p \\, d\\mu(x) \u003c \\infty$ for some point $x_0 \\in M$.\nFor $\\mu, \\nu \\in \\mathcal{P}_p(M)$, a *coupling* of $\\mu$ and $\\nu$ is a Borel probability measure $\\gamma$ on $M \\times M$ whose [[marginals|0MXT6PRZZ6]] are $\\mu$ and $\\nu$.\nThat is, $\\gamma(A \\times M) = \\mu(A)$ and $\\gamma(M \\times A) = \\nu(A)$ for every Borel set $A \\subseteq M$.\nLet $\\Gamma(\\mu, \\nu)$ be the set of couplings.\nThe *$p$-Wasserstein distance* is\n$$W_p(\\mu, \\nu) = \\left( \\inf_{\\gamma \\in \\Gamma(\\mu, \\nu)} \\int_{M \\times M} d(x, y)^p \\, d\\gamma(x, y) \\right)^{1/p}.$$\n\n## Remarks\n\nThe condition on $\\mu$ does not depend on the choice of $x_0$, by the triangle inequality.\nThe product measure $\\mu \\otimes \\nu$ is always a coupling, so $\\Gamma(\\mu, \\nu)$ is not empty and $W_p(\\mu, \\nu) \u003c \\infty$.\nThe function $W_p$ is a metric on $\\mathcal{P}_p(M)$, and the infimum is attained.\n\nExample: let $\\delta_x$ be the [[Dirac measure|0ACRGYNHSR]] at $x$.\nThe only coupling of $\\delta_x$ and $\\delta_y$ is $\\delta_{(x, y)}$, so $W_p(\\delta_x, \\delta_y) = d(x, y)$.\n\nNon-example: on $M = \\R$, the standard Cauchy distribution has $\\int \\abs{x} \\, d\\mu = \\infty$.\nSo it does not lie in $\\mathcal{P}_1(\\R)$, and $W_1$ is not defined for it.\n\nOn $\\R$, with distribution functions $F_\\mu$ and $F_\\nu$, one has $W_1(\\mu, \\nu) = \\int_{-\\infty}^{\\infty} \\abs{F_\\mu(t) - F_\\nu(t)} \\, dt$.\nIn $\\mathcal{P}_p(M)$, $W_p(\\mu_n, \\mu) \\to 0$ if and only if $\\mu_n \\to \\mu$ weakly and the $p$-th moments converge.\n\n## Explanations\n\n### L1 human\n\nA coupling is a plan for moving the mass of $\\mu$ to the mass of $\\nu$.\nThe distance is the cost of the cheapest plan, where moving mass by $r$ costs $r^p$ per unit.\n",
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