For all probability distributions $q$ over $x$, the following is a function of the parameters $\theta$ of a probability distribution $p(y,x; \theta)$: $$ f(\theta)=\sum_x q(x) \log\frac{p(y,x ; \theta)}{q(x)}$$
I'm trying to assert that this is not necessarily concave (as a function of $\theta$) using a counter-example. Intuitively, I can imagine some probability distribution $p$ that has two peaks and one valley (like two adjacent normal distributions) and maybe simplify by having $q$ get probability $1$ for a single $x$ and there are no other possible $x$ (i.e., no sum needed?), but is there some concrete probability distribution $p$ that I could use? (a formula)?
Thanks