A Tight Lower Bound for the Hellinger Distance with Given Means and Variances

Understand

The binary divergences that are divergences between probability measures defined on the same 2-point set have an interesting property.

• For the chi-squared divergence and the relative entropy, it is known that their binary divergence attain lower bounds with given means and variances, respectively.
• In this note, we show that the binary divergence of the squared Hellinger distance has the same property and propose an open problem that what conditions are needed for f-divergence to satisfy this property.