We investigate two options for performing Bayesian inference on spatial log-Gaussian Cox processes: 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 INLA over a selection of 18 simulated scenarios. The results question the notion that INLA is both significantly faster and more robust than MCMC in this setting; 100,000 iterations of the MALA algorithm running in 17 minutes on a desktop PC delivered greater predictive accuracy than INLA, which ran in 5.5 minutes.
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