= Coupling characterization of total variation distance
{title2=$d_{\mathrm{TV}}(P,Q)=\min\mathbb P(X\ne Y)$}
A <coupling of probability distributions> $P,Q$ is a pair of <random variables> $(X,Y)$ with those <marginal distributions>. On a <finite set>, and more generally on a <standard Borel space>, the least achievable mismatch is
$$
d_{\mathrm{TV}}(P,Q)=\min\{\mathbb P(X\ne Y):(X,Y)\text{ is a coupling of }P,Q\}.
$$
A <maximal coupling> attains the minimum; maximal here refers to agreement. An arbitrary <coupling> can have a much larger mismatch.<Markov Chains and Mixing Times, second edition by David A. Levin and others>[§4.2, Example 4.6 and Proposition 4.7, printed pp. 49–52: finite-state coupling and common-mass construction]{ref}
For the four-outcome example, assign the following joint <probability mass>: $(A,A)$ gets $0.1$, $(B,B)$ gets $0.1$, $(C,C)$ gets $0.2$, and $(D,D)$ gets $0.3$. The remaining $0.3$ must have $X=A$; place $0.2$ on $(A,B)$ and $0.1$ on $(A,C)$. Row sums give $P$, column sums give $Q$, and the <diagonal> contains $0.7$. \b[Agreement can reach 70%; mismatch cannot fall below 30%.]
\Image[/total-variation-maximal-coupling.png]
{title=A <maximal coupling> of two four-outcome <probability distributions>}
{id=image-total-variation-maximal-coupling}
{description=The original joint-mass example retains all common <probability mass> on its diagonal. The two off-diagonal entries account for the minimum mismatch $0.3$.}
To see the lower bound, for any <event> $E$,
$$
|P(E)-Q(E)|=|\mathbb E(\mathbf1_E(X)-\mathbf1_E(Y))|\leq\mathbb P(X\ne Y).
$$
The two <indicator functions> agree whenever $X=Y$. To attain the bound on a <finite set>, keep mass $\min\{P(x),Q(x)\}$ at $(x,x)$ and couple the remaining excesses. The two residual <probability distributions> live on disjoint <subsets>, so their part necessarily mismatches. When the <probability distributions> are identical, the residual part disappears entirely.
This is a choice of joint construction, not a claim that independently drawn observations are usually equal. Two <independent random variables> with the same <continuous probability distribution> can disagree with <probability> one even though their <total variation distance> is zero. The construction of a <maximal coupling> deliberately introduces <statistical dependence> to reveal how similar the marginal laws are.
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