This paper introduces a new method for performing computational inference onlog-Gaussian Cox processes. The likelihood is approximated directly by makingnovel use of a continuously specified Gaussian random field. We show that forsufficiently smooth Gaussian random field prior distributions, theapproximation can converge with arbitrarily high order, while an approximationbased on a counting process on a partition of the domain only achievesfirst-order convergence. The given results improve on the general theory ofconvergence of the stochastic partial differential equation models, introducedby Lindgren et al. (2011). The new method is demonstrated on a standard pointpattern data set and two interesting extensions to the classical log-GaussianCox process framework are discussed. The first extension considers variablesampling effort throughout the observation window and implements the method ofChakraborty et al. (2011). The second extension constructs a log-Gaussian Coxprocess on the world's oceans. The analysis is performed using integratednested Laplace approximation for fast approximate inference.
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