For discrete random variables X,Y with total variation distance at most epsilon, the equivocation H(X|Y) satisfies the tight bound |H(X|Y)-H(X'|Y')| ≤ epsilon log(|X|-1) + h(epsilon).
Estimating mutual information via kolmogoro v distance,
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A Tight Uniform Continuity Bound for Equivocation
For discrete random variables X,Y with total variation distance at most epsilon, the equivocation H(X|Y) satisfies the tight bound |H(X|Y)-H(X'|Y')| ≤ epsilon log(|X|-1) + h(epsilon).