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$\phi $-divergences; $\phi $-informations; power divergences; power entropies; Shannon entropy; quadratic entropy; Bayes error; Simpson diversity; Emlen diversity
The paper summarizes and extends the theory of generalized $\phi $-entropies $H_{\phi }(X)$ of random variables $X$ obtained as $\phi $-informations $I_{\phi }(X;Y)$ about $X$ maximized over random variables $Y$. Among the new results is the proof of the fact that these entropies need not be concave functions of distributions $p_{X}$. An extended class of power entropies $H_{\alpha }(X)$ is introduced, parametrized by $\alpha \in {\mathbb{R}}$, where $H_{\alpha }(X)$ are concave in $p_{X}$ for $\alpha \ge 0$ and convex for $\alpha <0$. It is proved that all power entropies with $\alpha \le 2$ are maximal $\phi $-informations $I_{\phi }(X;X)$ for appropriate $\phi $ depending on $\alpha $. Prominent members of this subclass of power entropies are the Shannon entropy $H_{1}(X)$ and the quadratic entropy $H_{2}(X)$. The paper investigates also the tightness of practically important previously established relations between these two entropies and errors $e(X)$ of Bayesian decisions about possible realizations of $X$. The quadratic entropy is shown to provide estimates which are in average more than 100 % tighter those based on the Shannon entropy, and this tightness is shown to increase even further when $\alpha $ increases beyond $\alpha =2$. Finally, the paper studies various measures of statistical diversity and introduces a general measure of anisotony between them. This measure is numerically evaluated for the entropic measures of diversity $H_1(X)$ and $H_2(X)$.
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