Title: An Efficient Variational Method for Fitting Log-Gaussian Cox Processes
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Log-Gaussian Cox processes; Variational Gaussian Approximation; Voronoi Tessellation; Coordinate Ascent Maximization

Abstract: Log-Gaussian Cox Processes (LGCP) have been widely used for modeling spatial point patterns. However, fitting LGCP is computationally challenging due to a nested structure involving Poisson process and latent Gaussian random field. To address these issues, we first approximate the intractable LGCP likelihood  based on the Voronoi tessellation method. Then, using variational Gaussian approximation, we transform the problem of fitting LGCP into maximizing the evidence lower bound which admits an explicit expression. We design a novel coordinate ascent maximization algorithm which updates the parameter blocks by Newton method and fixed-point method, respectively. To further enhance the computational efficiency, we adopt a nearest neighbor Gaussian process as the prior for the latent Gaussian random field, and the cost of inverting large covariance matrices is greatly reduced via the Woodbury formula. Theoretically, we prove the existence and uniqueness of the optimal solution to the strongly concave objective function, and the convergence of the proposed algorithm is established. Numerical results demonstrate the computational and inferential benefits of our method in modeling log-intensity surface over competing methods.

TL;DR: We introduce VoGCAM, a novel approach for fitting log-Gaussian Cox processes.