Exercise 15.12: $f$-Divergences

chapter 15

(a)

Obvious.

(b)

Reverse KL divergence.

(c)

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

(d)

$$\begin{array}{}\text{(1)}& \int q-q\sqrt{\frac{p}{q}}\text{d}\nu =1-\int \sqrt{pq}\text{d}\nu =\frac{1}{2}{H}^2(\mathbb{P},\mathbb{Q})\end{array}$$