  The log Gaussian Cox process is a flexible class of point pattern models for
capturing spatial and spatio-temporal dependence for point patterns. Model
fitting requires approximation of stochastic integrals which is implemented
through discretization of the domain of interest. With fine scale
discretization, inference based on Markov chain Monte Carlo is computationally
heavy because of the cost of repeated iteration or inversion or Cholesky
decomposition (cubic order) of high dimensional covariance matrices associated
with latent Gaussian variables. Furthermore, hyperparameters for latent
Gaussian variables have strong dependence with sampled latent Gaussian
variables. Altogether, standard Markov chain Monte Carlo strategies are
inefficient and not well behaved.
  In this paper, we propose an efficient computational strategy for fitting and
inferring with spatial log Gaussian Cox processes. The proposed algorithm is
based on a pseudo-marginal Markov chain Monte Carlo approach. We estimate an
approximate marginal posterior for parameters of log Gaussian Cox processes and
propose comprehensive model inference strategy. We provide details for all of
the above along with some simulation investigation for the univariate and
multivariate settings. As an example, we present an analysis of a point pattern
of locations of three tree species, exhibiting positive and negative
interaction between different species.
