[Paper Review] Posterior consistency for $n$ in the binomial $(n,p)$ problem with both parameters unknown - with applications to quantitative nanoscopy

Laura Fee Schneider, Johannes Schmidt-Hieber+4 more|arXiv (Cornell University)|Sep 7, 2018

Statistical Methods and Inference50 references4 citations

TL;DR

This paper establishes posterior consistency and a Bernstein-von Mises theorem for estimating the binomial population size $n$ when both $n$ and $p$ are unknown, under the regime where $p \to 0$ and $n \to \infty$ as $k \to \infty$. It introduces a new class of Bayesian estimators for $n$ and demonstrates their strong performance in simulations and a real-world application in super-resolution 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 $p ightarrow0$ and $n ightarrow\infty$ as $k o\infty$. 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.

Motivation & Objective

• To address the challenging problem of estimating the population size $n$ in binomial$(n,p)$ when both parameters are unknown and $p$ is small.
• To establish theoretical guarantees—posterior contraction and a Bernstein-von Mises type result—under asymptotic regimes where $p \to 0$ and $n \to \infty$ as the sample size $k \to \infty$.
• To develop a new class of Bayesian estimators for $n$ that are robust in low-$p$ regimes.
• To evaluate the performance of these estimators through comprehensive simulations and a real-world application in super-resolution microscopy.

Proposed method

• The authors consider a large class of priors on $n$ and $p$ and derive posterior contraction rates under the joint asymptotic regime $p \to 0$, $n \to \infty$, $k \to \infty$.
• They establish a Bernstein-von Mises type theorem, showing that the posterior distribution of $n$ concentrates around the true value at a parametric rate under regularity conditions.
• A novel class of objective Bayesian estimators is proposed, designed to adapt to the sparsity of observations when $p$ is small.
• The estimators are constructed using hierarchical priors that regularize the estimation of $n$ in high-dimensional, sparse settings.
• Theoretical results are supported by a simulation study comparing the new estimators to existing methods in terms of bias, variance, and mean squared error.
• The method is applied to real data from quantitative nanoscopy, where the number of detectable molecules ($n$) must be inferred from sparse, noisy imaging data.

Experimental results

Research questions

• RQ1 Under what conditions does the posterior distribution of $n$ contract to the true value when both $n$ and $p$ are unknown and $p \to 0$ as $k \to \infty$?
• RQ2 Can a Bernstein-von Mises theorem be established for the binomial$(n,p)$ model in the sparse regime where $p \to 0$ and $n \to \infty$?
• RQ3 How do the proposed Bayesian estimators for $n$ compare to existing methods in terms of frequentist performance under low-$p$ conditions?
• RQ4 What is the empirical performance of the new estimators in real-world super-resolution microscopy data?

Key findings

• Posterior contraction for $n$ is established under a broad class of priors when $p \to 0$ and $n \to \infty$ as $k \to \infty$, indicating reliable concentration of the posterior around the true $n$.
• A Bernstein-von Mises type theorem holds in the sparse regime, implying that the posterior distribution of $n$ is asymptotically normal with variance matching the inverse Fisher information.
• The proposed Bayesian estimators demonstrate superior performance in simulations, particularly in low-$p$ settings where frequentist estimators often fail.
• The new estimators show robustness and accuracy in a real application to super-resolution microscopy, where they successfully recover the number of fluorescent molecules from sparse imaging data.
• Simulation results confirm that the estimators achieve low bias and mean squared error even when $p$ is very small and $n$ is large.
• The application to nanoscopy demonstrates the practical utility of the Bayesian framework in quantitative biological imaging.