Exercise 15.12: $f$-Divergences

chapter 15

(a)

Obvious.

(b)

Reverse KL divergence.

(c)

Use $(\sqrt{p} - \sqrt{q})^2 = q \bigl( \sqrt{\tfrac{p}{q}} - 1 \bigr)^2$.

(d)

\begin{align} \int \! q - q \sqrt{\tfrac{p}{q}} \, \sd \nu = 1 - \int \! \sqrt{pq} \, \sd \nu = \frac{1}{2} H^2 (\P , \Q) \end{align}