Neural Methods for Multiple Systems Estimation Models

TL;DR

This work tackles the estimation of hidden population sizes via Multiple Systems Estimation under data censoring and resource constraints. It introduces simulation-based amortized Bayesian methods, Neural Bayes Estimators (NBE) for posterior medians and Neural Posterior Estimators (NPE) for full posteriors, trained on synthetic data from the MSE model with Poisson counts and log-linear structure. Across extensive simulations and two real datasets (Modern Slavery in the UK and Female Drug Users in North East England), NBEs and NPEs achieve accuracy and uncertainty quantification competitive with or superior to MCMC, while offering orders-of-magnitude faster inference and robustness to convergence failures, particularly in sparse settings. The results show NPE provides well-calibrated posteriors and credible predictive checks, while NBE serves as a fast screening tool; limitations include fixed priors, lack of formal model selection, and the need for high-quality simulations to cover real-world scenarios.

Abstract

Estimating the size of hidden populations using Multiple Systems Estimation (MSE) is a critical task in quantitative sociology; however, practical application is often hindered by imperfect administrative data and computational constraints. Real-world datasets frequently suffer from censoring and missingness due to privacy concerns, while standard inference methods, such as Maximum Likelihood Estimation (MLE) and Markov chain Monte Carlo (MCMC), can become computationally intractable or fail to converge when data are sparse. To address these limitations, we propose a novel simulation-based Bayesian inference framework utilizing Neural Bayes Estimators (NBE) and Neural Posterior Estimators (NPE). These neural methods are amortized: once trained, they provide instantaneous, computationally efficient posterior estimates, making them ideal for use in secure research environments where computational resources are limited. Through extensive simulation studies, we demonstrate that neural estimators achieve accuracy comparable to MCMC while being orders of magnitude faster and robust to the convergence failures that plague traditional samplers in sparse settings. We demonstrate our method on two real-world cases estimating the prevalence of modern slavery in the UK and female drug use in North East England.

Figures (15)

• Figure 1: Log Absolute Percentage Error (Log APE) as a function of true hidden population size for Neural Bayes Estimators (NBE) and Neural Posterior Estimates (NPE). Trend lines indicate the smoothed average performance for each estimator.
• Figure 2: Comparison of NBE and NPE performance across varying list sizes. The panels display the Log APE for Neural Bayes Estimators (left) and Neural Posterior Estimates (right) calculated from the test dataset estimates (n=10,000).
• Figure 3: Comparison of NBE and NPE performance across varying number of neurons. The panels display the Log APE for Neural Bayes Estimators (left) and Neural Posterior Estimates (right) calculated from the test dataset estimates (n=10,000).
• Figure 4: Comparison of NBE and NPE performance across varying number of hidden layers. The panels display the Log APE for Neural Bayes Estimators (left) and Neural Posterior Estimates (right) calculated from the test dataset estimates (n=10,000).
• Figure 5: Comparison of NBE and NPE performance across varying upper censoring level. The panels display the Log APE for Neural Bayes Estimators (left) and Neural Posterior Estimates (right) calculated from the test dataset estimates (n=10,000).