arXiv subjects
Publications and source records attributed to Gustavo L. Gilardoni.
It is shown that the $f$-divergence between two probability measures $P$ and $R$ equals the supremum of the same $f$-divergence computed over all finite measurable partitions of the original space, thus generalizing results previously proved by Gel'fand and Yaglom and by Peres for the Information Divergence and more recently by Dukkipati, Bhatnagar and Murty for the Tsallis' and Renyi's divergences.
We study conditions on $f$ under which an $f$-divergence $D_f$ will satisfy $D_f \geq c_f V^2$ or $D_f \geq c_{2,f} V^2 + c_{4,f} V^4$, where $V$ denotes variational distance and the coefficients $c_f$, $c_{2,f}$ and $c_{4,f}$ are {\em best possible}. As a consequence, we obtain lower bounds in terms of $V$ for many well known distance and divergence measures. For instance, let $D_{(α)} (P,Q) = [α(α-1)]^{-1} [\int q^α p^{1-α} d μ-1]$ and ${\cal I}_α(P,Q) = (α-1)^{-1} \log [\int p^αq^{1-α} d μ]$ be respectively the {\em relative information of type} ($1-α$) and {\em Rényi's information gain of order} $α$. We show that $D_{(α)} \geq {1/2} V^2 + {1/72} (α+1)(2-α) V^4$ whenever $-1 \leq α\leq 2$, $α\not= 0,1$ and that ${\cal I}_α = \fracα{2} V^2 + {1/36} α(1 + 5 α- 5 α^2) V^4$ for $0 < α< 1$. Pinsker's inequality $D \geq {1/2} V^2$ and its extension $D \geq {1/2} V^2 + {1/36} V^4$ are special cases of each one of these.