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

Abstract

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 $\mathit{p}\to 0$ and $\mathit{n}\to \infty$ as $\mathit{k}\to \infty$. 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.