Multiple systems estimation is a key approach for quantifying hidden populations such as the number of victims of modern slavery. The UK Government published an estimate of 10,000 to 13,000 victims, constructed by the present author, as part of the strategy leading to the Modern Slavery Act 2015. This estimate was obtained by a stepwise multiple systems method based on six lists. Further investigation shows that a small proportion of the possible models give rather different answers, and that other model fitting approaches may choose one of these. Three data sets collected in the field of modern slavery, together with a data set about the death toll in the Kosovo conflict, are used to investigate the stability and robustness of various multiple systems estimate methods. The crucial aspect is the way that interactions between lists are modelled, because these can substantially affect the results. Model selection and Bayesian approaches are considered in detail, in particular to assess their stability and robustness when applied to real modern slavery data. A new Markov Chain Monte Carlo Bayesian approach is developed; overall, this gives robust and stable results at least for the examples considered. The software and datasets are freely and publicly available to facilitate wider implementation and further research.
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