Going off grid: Computationally efficient inference for log-Gaussian Cox processes

This paper introduces a new method for performing computational inference on log-Gaussian Cox processes. The likelihood is approximated directly by making novel use of a continuously specified Gaussian random field. We show that for sufficiently smooth Gaussian random field prior distributions, the approximation can converge with arbitrarily high order, while an approximation based on a counting process on a partition of the domain only achieves first-order convergence. The given results improve on the general theory of convergence of the stochastic partial differential equation models, introduced by Lindgren et al. (2011). The new method is demonstrated on a standard point pattern data set and two interesting extensions to the classical log-Gaussian Cox process framework are discussed. The first extension considers variable sampling effort throughout the observation window and implements the method of Chakraborty et al. (2011). The second extension constructs a log-Gaussian Cox process on the world’s oceans. The analysis is performed using integrated nested Laplace approximation for fast approximate inference.

💡 Research Summary

This paper introduces a novel computational framework for Bayesian inference on log‑Gaussian Cox processes (LGCPs) that avoids the traditional reliance on a regular lattice to approximate the latent intensity field. The authors propose to represent the latent Gaussian random field Z(s) as a finite‑dimensional, continuously specified expansion
 Z(s)=∑_{i=1}^n z_i φ_i(s),
where the coefficients z are multivariate normal and the basis functions φ_i are compactly supported piecewise‑linear functions defined on a triangulated mesh. This construction is essentially the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011), which yields a Gaussian Markov random field (GMRF) with a sparse precision matrix while allowing arbitrary mesh geometry and local refinement.

The key methodological advance lies in the treatment of the intractable LGCP likelihood
 π(y|Z)=exp{−∫_Ω exp