  The log Gaussian Cox process is a flexible class of Cox processes, whose
intensity surface is stochastic, for incorporating complex spatial and time
structure of point patterns. The straightforward inference based on Markov
chain Monte Carlo is computationally heavy because the computational cost of
inverse or Cholesky decomposition of high dimensional covariance matrices of
Gaussian latent variables is cubic order of their dimension. Furthermore, since
hyperparameters for Gaussian latent variables have high correlations with
sampled Gaussian latent processes themselves, standard Markov chain Monte Carlo
strategies are inefficient. In this paper, we propose an efficient and scalable
computational strategy for spatial log Gaussian Cox processes. The proposed
algorithm is based on pseudo-marginal Markov chain Monte Carlo approach. Based
on this approach, we propose estimation of approximate marginal posterior for
parameters and comprehensive model validation strategies. We provide details
for all of the above along with some simulation investigation for univariate
and multivariate settings and analysis of a point pattern of tree data
exhibiting positive and negative interaction between different species.
