Posterior analysis of n in the binomial (n,p) problem with both parameters unknown—with applications to quantitative nanoscopy
Johannes Schmidt-Hieber, Laura Fee Schneider, Thomas Staudt, Andrea Krajina, Timo Aspelmeier, Axel Munk
The Annals of Statistics · 2021 · Volume 49 · Issue 6
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→0 and n→∞ as k→∞. 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.
Keywords
62F12, 62F15, 62G05, 62P10, 62P35, Bayesian estimation, Bernstein-von Mises theorem, beta-binomial likelihood, Binomial distribution, posterior contraction, quantitative cell imaging
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