= Coupling
{wiki=Coupling_(probability)}
= Coupling of probability distributions
{synonym}
= Probabilistic coupling
{synonym}
A <coupling of probability distributions> $\mu,\nu$ is a <joint probability distribution> of <random variables> $(X,Y)$ whose <marginal distributions> are $\mu,\nu$. It lets us choose the <statistical dependence> between two random constructions while preserving the individual laws. This choice can make a comparison much easier: giving two simulations the same random input may keep their paths close, even though separately generating both paths would hide that relationship.<Markov Chains and Mixing Times, second edition by David A. Levin and others>[§4.2, coupling definition and Example 4.6, printed pp. 49–50]{ref}
For example, two fair coin tosses can be generated independently, with mismatch <probability> $1/2$, or by tossing once and setting $Y=X$, with mismatch <probability> zero. Both constructions have the same two <Bernoulli distributions> as marginals. Their joint laws differ. A <maximal coupling> makes the values agree as often as possible; a <coalescing coupling> arranges permanent agreement of two processes after a random time. Minimizing expected geometric separation instead leads to <Wasserstein distance>.
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