Non-parametric calibration of multiple related radiocarbon determinations and their calendar age summarisation

Due to fluctuations in past radiocarbon (${}^{14}$C) levels, calibration is required to convert ${}^{14}$C determinations $X_i$ into calendar ages ${\theta }_i$. In many studies, we wish to calibrate a set of related samples taken from the same site or context, which have calendar ages drawn from the same shared, but unknown, density $f\left(\theta \right)$. Calibration of $X_1,\dots ,X_n$ can be improved significantly by incorporating the knowledge that the samples are related. Furthermore, summary estimates of the underlying shared $f\left(\theta \right)$ can provide valuable information on changes in population size/activity over time. Most current approaches require a parametric specification for $f\left(\theta \right)$ which is often not appropriate. We develop a rigorous non-parametric Bayesian approach using a Dirichlet process mixture model, with slice sampling to address the multi-modality typical within ${}^{14}$C calibration. Our approach simultaneously calibrates the set of ${}^{14}$C determinations and provides a predictive estimate for the underlying calendar age of a future sample. We show, in a simulation study, the improvement in calendar age estimation when jointly calibrating related samples using our approach, compared with calibration of each ${}^{14}$C determination independently. We also illustrate the use of the predictive calendar age estimate to provide insight on activity levels over time using three real-life case studies.