  Self-exciting Hawkes processes are used to model events which cluster in time
and space, and have been widely studied in seismology under the name of the
Epidemic Type Aftershock Sequence (ETAS) model. In the ETAS framework, the
occurrence of the mainshock earthquakes in a geographical region is assumed to
follow an inhomogeneous spatial point process, and aftershock events are then
modelled via a separate triggering kernel. Most previous studies of the ETAS
model have relied on point estimates of the model parameters due to the
complexity of the likelihood function, and the difficulty in estimating an
appropriate mainshock distribution. In order to take estimation uncertainty
into account, we instead propose a fully Bayesian formulation of the ETAS model
which uses a nonparametric Dirichlet process mixture prior to capture the
spatial mainshock process. Direct inference for the resulting model is
problematic due to the strong correlation of the parameters for the mainshock
and triggering processes, so we instead use an auxiliary latent variable
routine to perform efficient inference.
