Scalable Inference for Nonparametric Hawkes Process Using Pólya-Gamma Augmentation
This work addresses scalable inference for event data modeling, but it is incremental as it builds on existing nonparametric Hawkes process methods with algorithmic improvements.
The paper tackles the challenge of scalable inference for nonparametric Hawkes processes by modeling baseline intensity and triggering kernels with Gaussian processes and using Pólya-Gamma augmentation to enable efficient conjugate analytical inference. It develops EM and variational inference algorithms, demonstrating recovery of underlying characteristics on real data.
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, Pólya-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.