  Paleoclimate reconstruction on the Common Era (1-2000AD) provide critical
context for recent warming trends. This work leverages integrated nested
Laplace approximations (INLA) to conduct inference under a Bayesian
hierarchical model using data from three sources: a state-of-the-art prox
database (PAGES 2k), surface temperature observations (HadCRUT4), and latest
estimates of external forcings. INLA's computational efficiency allows to
explore several model formulations (with or without forcings, explicitly
modeling internal variability or not), as well as five data reduction
techniques. Two different validation exercises find a small impact of data
reduction choices, but a large impact for model choice, with best results for
the two models that incorporate external forcings. These models confirm that
man-made greenhouse gas emissions are the largest contributor to temperature
variability over the Common Era, followed by volcanic forcing. Solar effects
are indistinguishable from zero. INLA provide an efficient way to estimate the
posterior mean, comparable with the much costlier Monte Carlo Markov Chain
procedure, but with wider uncertainty bounds. We recommend using it for
exploration of model designs, but full MCMC solutions should be used for proper
uncertainty quantification.
