Hellinger distance

The Hellinger distance compares the square roots of the densities:

It is symmetric, satisfies the triangle inequality, and lies in $[0,1]$: 0 for identical distributions, 1 for distributions with disjoint support. The squared Hellinger distance is an f-divergence with $f(t)=(\sqrt{t}-1{)}^2$ (up to a constant).

Taking square roots softens the effect of large probability ratios, so Hellinger is more stable than KL where one density is near zero. It is related to TV by $H^2\le \delta \le \sqrt{2}H$.