Approximation of bayesian Hawkes process models with Inlabru

Hawkes process are very popular mathematical tools for modelling phenomena exhibiting a self-exciting behaviour. Typical examples are earthquakes occurrence, wild-fires, crime violence, trade exchange, and social network activity. The widespread use of Hawkes process in different fields calls for fast, reproducible, reliable, easy-to-code techniques to implement such models. We offer a technique to perform approximate Bayesian inference of Hawkes process parameters based on the use of the R-package Inlabru. Inlabru, in turn, relies on the INLA methodology to approximate the posterior of the parameters. The approximation is based on a decomposition of the Hakwes process likelihood in three parts, which are linearly approximated separately. The linear approximation is performed with respect to the mode of the posterior parameters distribution, which is determined with an iterative gradient-based method. The approximation of the posterior parameters is therefore deterministic, ensuring full reproducibility of the results. The proposed technique only required the user to provide the functions to calculate the different parts of the decomposed likelihood, while the optimization is performed through the R-package Inlabru. The limitations of this approach include the functional form of the different likelihood parts, which needs to be as linear as possible with respect to the parameters of the model. Moreover, care should be taken of the numerical stability of the provided functions.