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

Abstract

Estimation of the population size $ from $ i.i.d.\ binomial observations with unknown success probability $ is relevant to a multitude of applications and has a long history. Without additional prior information this is a notoriously difficult task when $ 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 \rightarrow0$ and \rightarrow\infty$ as \to\infty$. Furthermore, we suggest a new class of Bayesian estimators for $ 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.