Jan 07, 2010
Variational distance is the only metric f-divergence
Post @ 22:40:41 | f-divergence
A quite interesting property is that the variational divergence (a f-divergence) and its multiples are the only metric f-divergences. This was recently shown
Confliction of the Convexity and Metric Properties in f-Divergences (IEICE 2007).
If instead, we assume f strictly positive concave (and self-dual f(x)=x f(1/x)) then the formula of f-divergences yields metrics.
Frank.
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Re: Variational distance is the only metric f-divergence
It is interesting that square root of several f-divergences including Hellinger, Jensen-Shannon, etc are metric (see Hein and Bousquet (Hilbertian Metrics and Positive Definite Kernels on Probability Measures, AISTATS 2005) for example).
From : Memming @ 2010-01-10 05:34:29 Edit