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  "body": "\n## Summary\n\nA coupling of two [[probability measures|059TJN0NZ1]] $\\mu$ and $\\nu$ is a joint law of a pair $(X, Y)$.\nIn it, $X$ has law $\\mu$ and $Y$ has law $\\nu$.\nThe two marginals are fixed, but the dependence between $X$ and $Y$ is free to choose.\nA clever choice turns a comparison of two laws into a comparison of two random variables on one space.\nCouplings are a main tool for bounds in total variation distance, for convergence of [[Markov chains|0G48PYM96Q]], for [[stochastic ordering|HJPPRZ29YE]] and in optimal transport.\n\n## Explanations\n\n### L1 human\n\nTwo dice are a coupling of two uniform laws on $\\set{1, \\dots, 6}$; they can be thrown independently, or glued so that both always show the same face.\nBoth choices give the right law for each die, but very different pairs.\nTake a coin with heads probability $p$ and one with $q \\ge p$, and one uniform number $U$ in $[0, 1]$.\nSay that coin one shows heads when $U \u003c p$ and coin two when $U \u003c q$.\nThen each coin has the right law, and coin two shows heads whenever coin one does, which proves at once that coin two is more likely to show heads.\n",
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