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

The 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. In this paper we show posterior contraction as $k\to\infty$ in a setting where $p\rightarrow0$ and $n\rightarrow\infty$. The result holds for a large class of priors on $n$ which do not decay too fast. This covers several known Bayes estimators as well as a new class of estimators, which is governed by a scale parameter. We provide a comprehensive comparison of these estimators in a simulation study and extent their scope of applicability to a novel application from super-resolution cell microscopy.