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

TL;DR

This paper develops Bayesian methods for estimating the unknown population size and success probability in binomial models, especially when both parameters are small or large, with applications to super-resolution microscopy.

Contribution

It introduces new Bayesian estimators for $n$ and $p$, proves their theoretical properties, and demonstrates their effectiveness through simulations and real microscopy data.

Findings

Bayesian estimators outperform traditional methods in small $p$ scenarios.

Posterior contraction and Bernstein-von Mises results established for $p o 0$, $n o \infty$.

Application to super-resolution microscopy shows practical advantages.

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 $p\rightarrow0$ 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.