A Semiparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior

Hawkes processes are used in settings where past events increase the likelihood of future events occurring, resulting in a natural clustering structure. Traditional Hawkes process models treat events as occurring in continuous time, but in many applications only the number of events occurring within a sequence of time bins is observed. We propose the Gaussian Process Discrete Hawkes Process (GP-DHP), a semiparametric model for discrete-time self-exciting count data that places Gaussian-process priors on both the baseline and the excitation. Marginalizing the two GP components induces a single latent Gaussian trajectory. A finite-rank factorization of this collapsed prior permits maximum a posteriori (MAP) estimation without forming or factorizing a T\times T covariance matrix. External covariates can enter the baseline through the same construction. In simulations, GP-DHP recovers diverse excitation shapes and evolving baselines. In applications to weekly disease surveillance (Singapore dengue and German cryptosporidiosis), daily shooting counts (New York City and the Gun Violence Archive), and a daily worldwide terrorism-incident series, it attains the best held-out predictive accuracy on four of the five series and, on the fifth, an accuracy not significantly different from the best.