= Maximal coupling
A <maximal coupling> of two <probability distributions> maximizes the agreement <probability> $\mathbb P(X=Y)$ among all their <couplings>. Equivalently, it minimizes $\mathbb P(X\ne Y)$. The <coupling characterization of total variation distance> identifies the minimum mismatch with the <total variation distance> of the two marginal laws. The term maximal concerns agreement, rather than the size of a <correlation coefficient> or closeness in a geometric <metric>.
On a <finite set>, retain all common <probability mass> at equal pairs and match the excesses elsewhere. The explicit four-outcome example in <coupling characterization of total variation distance> realizes this construction and checks its <marginal distributions>. A <maximal coupling> need not be unique: different joint arrangements of the residual mass can have the same optimal mismatch.
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