Scalable Inference for Nonparametric Hawkes Process Using P\'{o}lya-Gamma Augmentation

TL;DR

This paper introduces a scalable inference method for a nonparametric Hawkes process model using Pólya-Gamma augmentation, enabling efficient Bayesian inference and MAP estimation.

Contribution

It develops an EM and a variational inference algorithm for the sigmoid Gaussian Hawkes process, improving inference efficiency and accuracy.

Findings

Algorithms recover underlying triggering characteristics effectively.

Methods perform well on both simulated and real data.

Inference is computationally efficient for large datasets.

Abstract

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.