  Hawkes processes are a popular means of modeling the event times of
self-exciting phenomena, such as earthquake strikes or tweets on a topical
subject. Classically, these models are fit to historical event time data via
likelihood maximization. However, in many scenarios, the exact times of
historical events are not recorded for either privacy (e.g., patient admittance
to hospitals) or technical limitations (e.g., most transport data records the
volume of vehicles passing loop detectors but not the individual times). The
interval-censored setting denotes when only the aggregate counts of events at
specific time intervals are observed. Fitting the parameters of
interval-censored Hawkes processes requires designing new training objectives
that do not rely on the exact event times. In this paper, we propose a model to
estimate the parameters of a Hawkes process in interval-censored settings. Our
model builds upon the existing Hawkes Intensity Process (HIP) of in several
important directions. First, we observe that while HIP is formulated in terms
of expected intensities, it is more natural to work instead with expected
counts; further, one can express the latter as the solution to an integral
equation closely related to the defining equation of HIP. Second, we show how a
non-homogeneous Poisson approximation to the Hawkes process admits a tractable
likelihood in the interval-censored setting; this approximation recovers the
original HIP objective as a special case, and allows for the use of a broader
class of Bregman divergences as loss function. Third, we explicate how to
compute a tighter approximation to the ground truth in the likelihood. Finally,
we show how our model can incorporate information about varying interval
lengths. Experiments on synthetic and real-world data confirm our HIPPer model
outperforms HIP and several other baselines on the task of interval-censored
inference.
