  We investigate two options for performing Bayesian inference on spatial
log-Gaussian Cox processes assuming a spatially continuous latent field: Markov
chain Monte Carlo (MCMC) and the integrated nested Laplace approximation
(INLA). We first describe the device of approximating a spatially continuous
Gaussian field by a Gaussian Markov random field on a discrete lattice, and
present a simulation study showing that, with careful choice of parameter
values, small neighbourhood sizes can give excellent approximations. We then
introduce the spatial log-Gaussian Cox process and describe MCMC and INLA
methods for spatial prediction within this model class. We report the results
of a simulation study in which we compare MALA and the technique of
approximating the continuous latent field by a discrete one, followed by
approximate Bayesian inference via INLA over a selection of 18 simulated
scenarios. The results question the notion that the latter technique is both
significantly faster and more robust than MCMC in this setting; 100,000
iterations of the MALA algorithm running in 20 minutes on a desktop PC
delivered greater predictive accuracy than the default \verb=INLA= strategy,
which ran in 4 minutes and gave comparative performance to the full Laplace
approximation which ran in 39 minutes.
