English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let D and V denote respectively Information Divergence and Total Variation Distance. Pinsker's and Vajda's inequalities are respectively D &#x2265; [ 1/ 2] V2 and D &#x2265; log[( 2+V)/( 2-V)] - [( 2V)/( 2+V)]. In this paper, several generalizations and improvements of these inequalities are established for wide classes of &lt;;i&gt;f&lt;;/i&gt;-divergences. First, conditions on f are determined under which an f-divergence Df will satisfy Df &#x2265; cf V2 or Df &#x2265; c2,f V2 + c4,f V4, where the constants cf, c2,f and c4,f are best possible. As a consequence, lower bounds in terms of V are obtained for many well known distance and divergence measures, including the &#x03C7;2 and Hellinger's discrimination and the families of Tsallis' and Re&#x0301;nyi's divergences. For instance, if D(&#x03B1;) (P||Q) = [&#x03B1;(&#x03B1;-1)]-1 [&#x222B;p&#x03B1;q1-&#x03B1;d&#x03BC;-1] and &#x2111;&#x03B1; (P||Q) = (&#x03B1;-1)-1 log[&#x222B;p&#x03B1;q1-&#x03B1;d&#x03BC;] are respectively the relative information of type &#x03B1; and the Re&#x0301;nyi's information gain of order &#x03B1;, it is shown that D(&#x03B1;) &#x2265; [ 1/ 2] V2 + [ 1/ 72] (&#x03B1;+1)(2-&#x03B1;) V4 whenever -1 &#x2264; &#x03B1; &#x2264; 2, &#x03B1; &#x2260; 0,1 and that &#x2111;&#x03B1; &#x2265; [( &#x03B1;)/ 2] V2 + [ 1/ 36] &#x03B1;(1 + 5 &#x03B1;- 5 &#x03B1;2 ) V4 for 0 &lt;; &#x03B1; &lt;; 1. In a somewhat different direction, and motivated by the fact that these Pinsker's type lower bounds are accurate only for small variation (V close to zero), lower bounds for Df which are accurate for both small and large variation (V close to two) are also obtained. In the special case of the information divergence they imply that D &#x2265; log[ 2/( 2-V)] - [( 2-V)/2] log[( 2+V)/2], which uniformly improves Vajda's inequality.

On Pinsker's and Vajda's Type Inequalities for Csiszár's $f$ -Divergences