Examining how past human populations responded to environmental and climatic changes is a central focus of the historical sciences. The use of summed probability distributions (SPD) of radiocarbon dates as a proxy for estimating relative population sizes provides a widely applicable method in this research area. Paleodemographic reconstructions and modeling with SPDs, however, are stymied by a lack of accepted methods for model fitting, tools for assessing the demographic impact of environmental or climatic variables, and a means for formal multi-model comparison. These deficiencies severely limit our ability to reliably resolve crucial questions of past human-environment interactions. We propose a solution using Approximate Bayesian Computation (ABC) to fit complex demographic models to observed SPDs. Using a case study from Rapa Nui (Easter Island), a location that has long been the focus of debate regarding the impact of environmental and climatic changes on its human population, we find that past populations were resilient to environmental and climatic challenges. Our findings support a growing body of evidence showing stable and sustainable communities on the island. The ABC framework offers a novel approach for exploring regions and time periods where questions of climate-induced demographic and cultural change remain unresolved.