The $\alpha$-divergences include the well-known Kullback-Leibler divergence, Hellinger distance and $\chi^2$-divergence. In this paper, we derive differential and integral relations between the $\alpha$-divergences that are generalizations of the relation between the Kullback-Leibler divergence and the $\chi^2$-divergence. We also show tight lower bounds for the $\alpha$-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：$\ alpha$- divegence 包括众所周知的 Kullback- Leverer 差异、 Hellinger 距离和 $\chi ⁇ 2$- divegence 。 在本文中,我们在 Kullback- Leiber 差异和 $\ chi ⁇ 2$- divegence 之间的关系上产生了差异和整体关系。 我们还显示出在给定手段和差异下 $\ alphaback- Lebel 差异的严格下限。 特别是, 我们显示出一个必要和充分的条件, 即二进制差异, 即同一两分点的概率计量之间的差异, 总是达到较低的界限。 Kullback- Leber 差异、 Hellinger 距离和 $\ chi2$- divegence 满足了这一条件 。