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 $f(\theta)$. Calibration of $X_1, \ldots,

X_n$ can be improved significantly by incorporating the knowledge that the

samples are related. Furthermore, summary estimates of the underlying shared

$f(\theta)$ can provide valuable information on changes in population

size/activity over time. Most current approaches require a parametric

specification for $f(\theta)$ 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 multimodality 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.