Approximation of Bayesian Hawkes process with inlabru
Statistics Seminar
29th November 2024, 1:00 pm – 2:00 pm
Fry Building, 2.04
Hawkes process are very popular mathematical tools for modeling phenomena exhibiting a self-exciting or self-correcting behavior. 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. We provide an application to earthquake data from Italy in which we compare the results obtained with inlabru with results obtained used MCMC through the bayesianETAS R-package. We then show a spatio-temporal extension of the model, as well as a model formulation to introduce covariates in modelling the number of offsprings. We conclude showing a more efficient modification of the proposed methodology relying on the EM algorithm to determine the mode.
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