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Regular version of the site

Article

The Minimum Increment of f-Divergences Given Total Variation Distances

Mathematical Methods of Statistics. 2016. Vol. 25. No. 4. P. 304-312.
Let (P_i,Q_i), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω_i,F_i) respectively. Assume that the pair (P_1,Q_1) is more informative than (P_0,Q_0) for testing problems. This amounts to say that I_f (P_1,Q_1) ≥ I_f (P_0,Q_0), where I_f (·, ·) is an arbitrary fdivergence. We find a precise lower bound for the increment of f-divergences I_f (P_1,Q_1) − I_f (P_0,Q_0) provided that the total variation distances ||Q_1 − P_1|| and ||Q_0 − P_0|| are given. This optimization problem can be reduced to the case where P_1 and Q_1 are defined on the space consisting of four points, and P_0 and Q_0 are obtained from P_1 and Q_1 respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f (P,Q) given ||Q − P||.