  The Hawkes process, a self-exciting point process, has a wide range of
applications in modeling earthquakes, social networks and stock markets. The
established estimation process requires that researchers have access to the
exact time stamps and spatial information. However, available data are often
rounded or aggregated. We develop a Bayesian estimation procedure for the
parameters of a Hawkes process based on aggregated data. Our approach is
developed for temporal, spatio-temporal, and mutually exciting Hawkes processes
where data are available over discrete time periods and regions. We show
theoretically that the parameters of the Hawkes process are identifiable from
aggregated data under general specifications. We demonstrate the method on
simulated temporal and spatio-temporal data with various model specifications
in the presence of one or more interacting processes, and under varying
coarseness of data aggregation. Finally, we examine the internal and
cross-excitation effects of airstrikes and insurgent violence events from
February 2007 to June 2008, with some data aggregated by day.
