  The Log-Gaussian Cox Process is a commonly used model for the analysis of
spatial point patterns. Fitting this model is difficult because of its
doubly-stochastic property, i.e., it is an hierarchical combination of a
Poisson process at the first level and a Gaussian Process at the second level.
Different methods have been proposed to estimate such a process, including
traditional likelihood-based approaches as well as Bayesian methods. We focus
here on Bayesian methods and several approaches that have been considered for
model fitting within this framework, including Hamiltonian Monte Carlo, the
Integrated nested Laplace approximation, and Variational Bayes. We consider
these approaches and make comparisons with respect to statistical and
computational efficiency. These comparisons are made through several
simulations studies as well as through applications examining both ecological
data and neuroimaging data.
