  The spatio-temporal Epidemic Type Aftershock Sequence (ETAS) model is widely
used to describe the self-exciting nature of earthquake occurrences. While
traditional inference methods provide only point estimates of the model
parameters, we aim at a full Bayesian treatment of model inference, allowing
naturally to incorporate prior knowledge and uncertainty quantification of the
resulting estimates. Therefore, we introduce a highly flexible, non-parametric
representation for the spatially varying ETAS background intensity through a
Gaussian process (GP) prior. Combined with classical triggering functions this
results in a new model formulation, namely the GP-ETAS model. We enable
tractable and efficient Gibbs sampling by deriving an augmented form of the
GP-ETAS inference problem. This novel sampling approach allows us to assess the
posterior 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 outperforms
standard models and thus demonstrate the predictive power for observed
earthquake catalogues including uncertainty quantification for the estimated
parameters. Finally, a case study for the l'Aquila region, Italy, with the
devastating event on 6 April 2009, is presented.
