f-Divergence

The f-divergence is defined as¹

$$ \operatorname{D}_f = \int f\left(\frac{p}{q}\right) q\mathrm d\mu, $$

where $p$ and $q$ are two densities and $\mu$ is a reference distribution.

For $f(x) = x \log x$ with $x=p/q$, f-divergence is reduced to the KL divergence

$$ \begin{align} &\int f\left(\frac{p}{q}\right) q\mathrm d\mu \\ =& \int \frac{p}{q} \log \left( \frac{p}{q} \right) \mathrm d\mu \\ =& \int p \log \left( \frac{p}{q} \right) \mathrm d\mu. \end{align} $$

For more special cases of f-divergence, please refer to wikipedia¹. Nowozin et al also provided a concise review of f-divergence².