  Self-exciting spatiotemporal Hawkes processes have found increasing use in
the study of large-scale public health threats ranging from gun violence and
earthquakes to wildfires and viral contagion. Whereas many such applications
feature locational uncertainty, i.e., the exact spatial positions of individual
events are unknown, most Hawkes model analyses to date have ignored spatial
coarsening present in the data. Three particular 21st century public health
crises -- urban gun violence, rural wildfires and global viral spread --
present qualitatively and quantitatively varying uncertainty regimes that
exhibit (a) different collective magnitudes of spatial coarsening, (b) uniform
and mixed magnitude coarsening, (c) differently shaped uncertainty regions and
-- less orthodox -- (d) locational data distributed within the `wrong'
effective space. We explicitly model such uncertainties in a Bayesian manner
and jointly infer unknown locations together with all parameters of a
reasonably flexible Hawkes model, obtaining results that are practically and
statistically distinct from those obtained while ignoring spatial coarsening.
This work also features two different secondary contributions: first, to
facilitate Bayesian inference of locations and background rate parameters, we
make a subtle yet crucial change to an established kernel-based rate model; and
second, to facilitate the same Bayesian inference at scale, we develop a
massively parallel implementation of the model's log-likelihood gradient with
respect to locations and thus avoid its quadratic computational cost in the
context of Hamiltonian Monte Carlo. Our examples involve thousands of
observations and allow us to demonstrate practicality at moderate scales.
