This work builds a novel point process and tools to use the Hawkes process with interval-censored data. Such data records the aggregated counts of events solely during specific time intervals -- such as the number of patients admitted to the hospital or the volume of vehicles passing traffic loop detectors -- and not the exact occurrence time of the events. First, we establish the Mean Behavior Poisson (MBP) process, a novel Poisson process with a direct parameter correspondence to the popular self-exciting Hawkes process. The event intensity function of the MBP is the expected intensity over all possible Hawkes realizations with the same parameter set. We fit MBP in the interval-censored setting using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter equivalence to uncover the parameters of the associated Hawkes process. Second, we introduce two novel exogenous functions to distinguish the exogenous from the endogenous events. We propose the multi-impulse exogenous function when the exogenous events are observed as event time and the latent homogeneous Poisson process exogenous function when the exogenous events are presented as interval-censored volumes. Third, we provide several approximation methods to estimate the intensity and compensator function of MBP when no analytical solution exists. Fourth and finally, we connect the interval-censored loss of MBP to a broader class of Bregman divergence-based functions. Using the connection, we show that the current state of the art in popularity estimation (Hawkes Intensity Process (HIP) (Rizoiu et al.,2017b)) is a particular case of the MBP process. We verify our models through empirical testing on synthetic data and real-world data. We find that on real-world datasets that our MBP process outperforms HIP for the task of popularity prediction.
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