EMbru: A Quick and Accurate Bayesian Inference Method for Hawkes Point Process Modeling

In this article, we introduce EMbru, a novel method to perform inference in spatio-temporal Hawkes models that combines a Bayesian version of Expectation-Maximization (EM) method with the Bayesian Integrated Nested Laplace Approximation (INLA) method, offering an advantage over EM by incorporating uncertainty into the estimates, but also a large computational time reduction over Markov Chain Monte Carlo methods. To assess the performance of EMbru, we conduct a simulation study comparing it with both the EM and inlabru implementations in terms of estimation accuracy and computational time. Additionally, to demonstrate its applicability, we analyze data on rat sightings in urban environments and earthquake data, incorporating a covariate into the background rate in the latter case to evaluate its flexibility and generalization capacity. The results indicate that EMbru is a robust and efficient alternative for Bayesian inference on spatio-temporal Hawkes processes.