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

Hawkes process models are used in settings where past events increase the likelihood of future events occurring. Many applications record events as counts on a regular grid, yet discrete-time Hawkes models remain comparatively underused and are often constrained by fixed-form baselines and excitation kernels. In particular, there is a lack of flexible, nonparametric treatments of both the baseline and the excitation in discrete time. To this end, we propose the Gaussian Process Discrete Hawkes Process (GP-DHP), a nonparametric framework that places Gaussian process priors on both the baseline and the excitation and performs inference through a collapsed latent representation. This yields smooth, data-adaptive structure without prespecifying trends, periodicities, or decay shapes, and enables maximum a posteriori (MAP) estimation with near-linear-time (O(T\log T)) complexity. A closed-form projection recovers interpretable baseline and excitation functions from the optimized latent trajectory. In simulations, GP-DHP recovers diverse excitation shapes and evolving baselines. In case studies on U.S. terrorism incidents and weekly Cryptosporidiosis counts, it improves test predictive log-likelihood over standard parametric discrete Hawkes baselines while capturing bursts, delays, and seasonal background variation. The results indicate that flexible discrete-time self-excitation can be achieved without sacrificing scalability or interpretability.

💡 Research Summary

The paper introduces the Gaussian Process Discrete Hawkes Process (GP‑DHP), a novel non‑parametric framework for modeling self‑exciting count data observed on a regular discrete time grid. Traditional discrete‑time Hawkes models typically impose restrictive parametric forms on the baseline intensity (often constant or simple sinusoidal) and on the excitation kernel (geometric, negative‑binomial, etc.). Such constraints limit the ability to capture long‑range memory, non‑stationarity, or evolving background trends that are common in applications like epidemiology, crime, or terrorism data.

GP‑DHP places independent Gaussian‑process (GP) priors on both the baseline function (b(t)) and the excitation kernel (f(d)). The baseline GP uses a composite kernel that combines a periodic RBF component (to capture seasonal cycles), a linear component (to capture long‑term trends), and a jitter term for numerical stability. The excitation GP is defined over discrete lags (d) and incorporates a non‑stationary construction: an amplitude envelope (a(d)=\sigma_f\exp(-\beta d^2)) and a warped input (g(d)=1-e^{-\beta d}\beta\ell_f). These are combined with a stationary RBF kernel on the warped inputs, yielding a covariance (K_f(d,d’) = a(d)a(d’)\exp{-\frac12