We derive inequalities of the form Δ (P, Q) ≤ H(P|R) + H(Q|R) which hold for every choice of probability measures P, Q, R, where $H(P|R)$ denotes the relative entropy of P with respect to R and Δ (P, Q) stands for a coupling type "distance" between P and Q. Using the chain rule for relative entropies and then specializing to Q with a given support we recover some of Talagrand's concentration of measure inequalities for product spaces.
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Amir Dembo (1997) studied this question.
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