Approximate Marginal Posterior for Log Gaussian Cox Processes

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.