  Spatio-temporal point process models play a central role in the analysis of
spatially distributed systems in several disciplines. Yet, scalable inference
remains computa- tionally challenging both due to the high resolution modelling
generally required and the analytically intractable likelihood function. Here,
we exploit the sparsity structure typical of (spatially) discretised
log-Gaussian Cox process models by using approximate message-passing
algorithms. The proposed algorithms scale well with the state dimension and the
length of the temporal horizon with moderate loss in distributional accuracy.
They hence provide a flexible and faster alternative to both non-linear
filtering-smoothing type algorithms and to approaches that implement the
Laplace method or expectation propagation on (block) sparse latent Gaussian
models. We infer the parameters of the latent Gaussian model using a structured
variational Bayes approach. We demonstrate the proposed framework on simulation
studies with both Gaussian and point-process observations and use it to
reconstruct the conflict intensity and dynamics in Afghanistan from the
WikiLeaks Afghan War Diary.
