Abstract

Estimating the size of hidden populations, such as victims of modern slavery or individuals engaging in illicit drug use, is a long-standing methodological challenge with significant implications for crime science and forensic statistics. Multiple Systems Estimation (MSE) is a widely adopted capture-recapture technique used to estimate these dark figures of crime by analysing incomplete, overlapping administrative lists. However, real-world forensic and sociological datasets are frequently imperfect, suffering from censoring and missingness due to stringent privacy concerns.

Operating at the intersection of computer science, mathematics, and social sciences, this paper proposes a simulation-based Bayesian inference framework utilising Neural Bayes Estimators (NBE) and Neural Posterior Estimators (NPE). By leveraging modern computational power, these neural networks bypass explicit likelihood calculations by learning to approximate posterior quantities from synthetic data generated by model simulators. Crucially, these neural methods are amortised: once the computationally intensive training is completed "offline," they provide instantaneous, highly efficient posterior estimates for any new dataset.

This "train once, infer many times" capability makes neural estimators ideal for secure or trusted research environments where sensitive crime data is stored but computational resources are limited. Through extensive simulation studies and real-world applications, including estimating the prevalence of modern slavery in the UK and female drug use in North East England, we demonstrate that neural estimators achieve accuracy comparable to MCMC while being orders of magnitude faster. Ultimately, this framework offers a robust, AI-driven solution to convergence failures in sparse settings, providing policymakers and justice systems with a powerful tool to reliably quantify hidden populations and inform evidence-based interventions.