Posterior analysis of $n$ in the binomial $(n,p)$ problem with both parameters unknown -- with applications to quantitative nanoscopy

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.