POSTERIOR ANALYSIS OF n IN THE BINOMIAL (n, p) PROBLEM WITH BOTH PARAMETERS UNKNOWN-WITH APPLICATIONS TO QUANTITATIVE NANOSCOPY

• 成果类型: Article
• 署名作者: Schmidt-Hieber, Johannes; Schneider, Laura Fee; Staudt, Thomas; Krajina, Andrea; Aspelmeier, Timo; Munk, Axel
• 署名单位: University of Twente; University of Gottingen; Max Planck Society; University of Gottingen; UNIVERSITY GOTTINGEN HOSPITAL
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2096
• 发表日期: 2021
• 页码: 3534-3558
• 关键词: population-size; localization microscopy; bayesian-estimation; cautionary note; estimators; FRAMEWORK; inference

摘要

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 -> infinity as k -> infinity. 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.