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The spatio-temporal Epidemic Type Aftershock Sequence (ETAS) model is widelyused to describe the self-exciting nature of earthquake occurrences. Whiletraditional inference methods provide only point estimates of the modelparameters, we aim at a full Bayesian treatment of model inference, allowingnaturally to incorporate prior knowledge and uncertainty quantification of theresulting estimates. Therefore, we introduce a highly flexible, non-parametricrepresentation for the spatially varying ETAS background intensity through aGaussian process (GP) prior. Combined with classical triggering functions thisresults in a new model formulation, namely the GP-ETAS model. We enabletractable and efficient Gibbs sampling by deriving an augmented form of theGP-ETAS inference problem. This novel sampling approach allows us to assess theposterior model variables conditioned on observed earthquake catalogues, i.e.,the spatial background intensity and the parameters of the triggering function.Empirical results on two synthetic data sets indicate that GP-ETAS outperformsstandard models and thus demonstrate the predictive power for observedearthquake catalogues including uncertainty quantification for the estimatedparameters. Finally, a case study for the l'Aquila region, Italy, with thedevastating event on 6 April 2009, is presented.

GP-ETAS: Semiparametric Bayesian inference for the spatio-temporal Epidemic Type Aftershock Sequence model