Posterior consistency for n in the binomial (n,p) problem with both parameters unknown - with applications to quantitative nanoscopy
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The 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. In this paper we show posterior contraction as k→∞ in a setting where p→0 and n→∞. The result holds for a large class of priors on n which do not decay too fast. This covers several known Bayes estimators as well as a new class of estimators, which is governed by a scale parameter. We provide a comprehensive comparison of these estimators in a simulation study and extent their scope of applicability to a novel application from super-resolution cell microscopy.
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