  We perform Bayesian inference on historical and late Holocene (last 2000
years) rates of sea-level change. The input data to our model are tide-gauge
measurements and proxy reconstructions from cores of coastal sediment. These
data are complicated by multiple sources of uncertainty, some of which arise as
part of the data collection exercise. Notably, the proxy reconstructions
include temporal uncertainty from dating of the sediment core using techniques
such as radiocarbon. The model we propose places a Gaussian process prior on
the rate of sea-level change, which is then integrated and set in an
errors-in-variables framework to take account of age uncertainty. The resulting
model captures the continuous and dynamic evolution of sea-level change with
full consideration of all sources of uncertainty. We demonstrate the
performance of our model using two real (and previously published) example data
sets. The global tide-gauge data set indicates that sea-level rise increased
from a rate with a posterior mean of 1.13 mm$/$yr in 1880 AD (0.89 to 1.28
mm$/$yr 95% credible interval for the posterior mean) to a posterior mean rate
of 1.92 mm$/$yr in 2009 AD (1.84 to 2.03 mm$/$yr 95% credible interval for the
posterior mean). The proxy reconstruction from North Carolina (USA) after
correction for land-level change shows the 2000 AD rate of rise to have a
posterior mean of 2.44 mm$/$yr (1.91 to 3.01 mm$/$yr 95% credible interval).
This is unprecedented in at least the last 2000 years.
