Approximation of bayesian Hawkes process models with Inlabru

TL;DR

This paper introduces a fast, reproducible, and reliable approximate Bayesian inference method for Hawkes process models using the R-package extit{inlabru} and INLA, significantly reducing computational time compared to MCMC-based methods.

Contribution

The authors develop a novel approximation technique for Bayesian inference of Hawkes processes leveraging extit{inlabru} and INLA, enhancing speed and reproducibility over existing MCMC approaches.

Findings

The method achieves similar accuracy to MCMC-based approaches.

It reduces computational time by a factor of 2 to 10.

Results are fully reproducible due to deterministic approximation.

Abstract

Hawkes process are very popular mathematical tools for modelling phenomena exhibiting a \textit{self-exciting} or \textit{self-correcting} behaviour. Typical examples are earthquakes occurrence, wild-fires, drought, capture-recapture, 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. The \inlabru R-package, in turn, relies on the INLA methodology to approximate the posterior of the parameters. Our Hawkes process approximation is based on a decomposition of the log-likelihood in three parts, which are linearly approximated separately. The linear approximation is performed with respect to the mode of the parameters' posterior 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 requires the user to provide the functions to calculate the different parts of the decomposed likelihood, which are internally linearly approximated by the R-package \inlabru. We provide a comparison with the \bayesianETAS R-package which is based on an MCMC method. The two techniques provide similar results but our approach requires two to ten times less computational time to converge, depending on the amount of data.