  In this paper, we consider the sigmoid Gaussian Hawkes process model: the
baseline intensity and triggering kernel of Hawkes process are both modeled as
the sigmoid transformation of random trajectories drawn from Gaussian processes
(GP). By introducing auxiliary latent random variables (branching structure,
P\'{o}lya-Gamma random variables and latent marked Poisson processes), the
likelihood is converted to two decoupled components with a Gaussian form which
allows for an efficient conjugate analytical inference. Using the augmented
likelihood, we derive an expectation-maximization (EM) algorithm to obtain the
maximum a posteriori (MAP) estimate. Furthermore, we extend the EM algorithm to
an efficient approximate Bayesian inference algorithm: mean-field variational
inference. We demonstrate the performance of two algorithms on simulated
fictitious data. Experiments on real data show that our proposed inference
algorithms can recover well the underlying prompting characteristics
efficiently.
