The $\alpha$-divergences include Kullback-Leibler divergence, Hellinger distance and $\chi^2$-divergence. We derive differntial and integral relations between $\alpha$-divergences that are generalizations of the relation between the Kullback-Leibler divergence and the $\chi^2$-divergence. We also show tight lower bounds for $\alpha$-divergences under given means and variances. In particular, we show a necessary and sufficient condition such that the binary divergences, which are divergences between probability measures on the same $2$-point set, always attain lower bounds. Kullback-Leibler divergence, Hellinger distance, and $\chi^2$-divergence satisfy this condition.
翻译：$\ alpha$- divegences 包括 Kullback- Leiber 差异、 Hellinger 距离和 $chi ⁇ 2$- divegences。 我们从 $\ alpha$- divegences 和 $\ chi ⁇ 2$- divegences 之间的差异和整体关系中产生差异和整体关系,这些关系是 Kullback- Leibel 差异和 $\ chi ⁇ 2$- divegences 之间的一般关系。 我们还显示,在给定手段和差异下, $\ alpha$- disgences 的下限非常低。 特别是, 我们显示出一个必要和充分的条件, 即二进制差异, 即同一 $- point 套置的概率计量方法之间的差异, 总是达到较低的界限。 Kullback- Lebel 差异、 Hellinger 距离和 $\\ 2$ diver- divences 满足了这一条件 。
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