  How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain?
We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound
which is tight up to a factor of $3/2$ for all other values of $m$. This
extends the prior results of Cox--Martin and Lv et al. on the analogous
question for even cycles. Our bounds result from a reduction to the following
maximum likelihood question: which probability mass $\mu$ on the edges of some
clique maximizes the probability that $m$ edges sampled independently from
$\mu$ form either a cycle or a path?
