Spatio-temporal hierarchical modeling is an extremely attractive way to model
the spread of crime or terrorism data over a given region, especially when the
observations are counts and must be modeled discretely. The spatio-temporal
diffusion is placed, as a matter of convenience, in the process model allowing
for straightforward estimation of the diffusion parameters through Bayesian
techniques. However, this method of modeling does not allow for the existence
of self-excitation, or a temporal data model dependency, that has been shown to
exist in criminal and terrorism data. In this manuscript we will use existing
theories on how violence spreads to create models that allow for both
spatio-temporal diffusion in the process model as well as temporal diffusion,
or self-excitation, in the data model. We will further demonstrate how Laplace
approximations similar to their use in Integrated Nested Laplace Approximation
can be used to quickly and accurately conduct inference of self-exciting
spatio-temporal models allowing practitioners a new way of fitting and
comparing multiple process models. We will illustrate this approach by fitting
a self-exciting spatio-temporal model to terrorism data in Iraq and demonstrate
how choice of process model leads to differing conclusions on the existence of
self-excitation in the data and differing conclusions on how violence is
spreading spatio-temporally
