Bayesian nonparametric marked Hawkes processes for earthquake modeling
We propose a Bayesian nonparametric model for marked Hawkes processes (MHPs). The conditional intensity function of the processes is decomposed into the ground process intensity and the mark density function. Our primary focus is on modeling the ground process intensity, while also providing a specifically defined model for the mark density function and its flexible alternative. The prior probability model for the intensity is carefully designed to balance flexibility with tractable posterior inference, achieved through a novel mixture modeling method. This model is motivated by seismology applications, where magnitude is treated as a mark associated with the time of earthquake occurrences. Accordingly, the mixture model basis is defined as a function of occurrence time and magnitude, with its functional form chosen to ensure model flexibility and alignment with earthquake data characteristics, such as the fact that larger magnitude earthquakes generate more subsequent shocks than smaller ones. The model is illustrated with three synthetic data examples and an earthquake occurrence data set.