  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.
