Let $x$ be a binary random variable and $z$ be an arbitrary random variable. $x$ and $z$ are, in general, not independent. Let $y_1, \ldots y_n$ be $n$ identically distributed binary random variables conditionally independent given $z$. A graphical model would have $x$ at the root, pointing to $z$, and $z$ pointing to each of the $y_i$. Consider random variable $c = \sum y_i$ and function $f(k) = p(x=1|c=k)$ defined from $k=0$ to $n$. How many local extrema can $f$ have, at most? That is, points where $(f(k)-f(k-1))(f(k+1)-f(k)) < 0$