  Estimation of the population size $n$ from $k$ i.i.d.\ binomial observations
with unknown success probability $p$ is relevant to a multitude of applications
and has a long history. Without additional prior information this is a
notoriously difficult task when $p$ becomes small, and the Bayesian approach
becomes particularly useful. For a large class of priors, we establish
posterior contraction and a Bernstein-von Mises type theorem in a setting where
$p\rightarrow0$ and $n\rightarrow\infty$ as $k\to\infty$. Furthermore, we
suggest a new class of Bayesian estimators for $n$ and provide a comprehensive
simulation study in which we investigate their performance. To showcase the
advantages of a Bayesian approach on real data, we also benchmark our
estimators in a novel application from super-resolution microscopy.
