I was asked to show that for two discrete RVs $X,Y$ with joint distribution $p\left(x,y\right)$ and marginal distributions $p\left(x\right)$ and $p\left(y\right)$, it holds that for any two distributions $r\left(x\right)$ and $s\left(y\right)$
$$I\left(X;Y\right)=\underset{r\left(x\right),s\left(y\right)}{\min}D_{KL}\left[p\left(x,y\right)\mid\mid r\left(x\right)\cdot s\left(y\right)\right]$$
Now,
$$I\left(X;Y\right)=D_{KL}\left[p\left(x,y\right)\mid\mid p\left(x\right)\cdot p\left(y\right)\right]=\sum_{x,y}p\left(x,y\right)\log\frac{p\left(x,y\right)}{p\left(x\right)p\left(y\right)}$$
so this amounts to showing $$\underset{r\left(x\right),s\left(y\right)}{\min}\sum_{x,y}p\left(x,y\right)\log\frac{p\left(x,y\right)}{r\left(x\right)s\left(y\right)}=\sum_{x,y}p\left(x,y\right)\log\frac{p\left(x,y\right)}{p\left(x\right)p\left(y\right)}$$
but how does one show this? Do I need to differentiate w.r.t. $r\left(x\right)$ and $s\left(y\right)$? Is that possible?