A Lower Bound on Arbitrary $f$--Divergences in Terms of the Total Variation

An important tool to quantify the likeness of two probability measures are f–divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among the f–divergences. It is shown that every f– divergence is bounded from below by a monotonous function of the total variation. Under appropriate regularity conditions, this function is shown to be monotonous. Remark: The proof of the main proposition is relatively easy, whence it is highly likely that the result is known. The author would be very grateful for any information regarding references or related work. 1 The total variation Let (Ω, σ) be a probability space. A signed measure ν is a σ–additive set function with values in R ∪ {−∞,∞}, and so that either ν > −∞ or ν < ∞. I will use the standard term measure if ν is nonnegative. To any signed measure ν, there corresponds a Hahn–Jordan decomposition of Ω into two measurable sets P,N so that P ∪N = Ω, P ∩N = ∅ and ν(.) = ν(. ∩ P ), ν(.) = −ν(. ∩N) (1) are both (nonnegative) measures. Obviously, ν = ν − ν. Furthermore, the representation ν(A) = sup B⊂A ν(B), ν(A) = − inf B⊂A ν(B) (2) holds for every measurable set A. For a proof of these facts see [2]. The measure 〈ν〉 = ν + ν is called the variation measure of ν, which in turn defines the email: [email protected]