Posterior concentration in spatio-temporal Hawkes processes

TL;DR

This paper introduces a Bayesian nonparametric approach for modeling and inferring spatio-temporal Hawkes processes, providing the first theoretical guarantees for such methods in this setting.

Contribution

It extends existing theoretical results from temporal to spatio-temporal Hawkes processes using Gaussian Process priors for a comprehensive Bayesian framework.

Findings

Derived explicit posterior contraction rates for the model

Generalized contraction results to the spatio-temporal setting

Provided the first theoretical guarantees for Bayesian nonparametrics in spatio-temporal point processes

Abstract

We develop a Bayesian nonparametric framework for inference in spatio-temporal Hawkes processes, extending existing theoretical results beyond the purely temporal setting. Our framework encompasses modelling both the background and triggering components of the Hawkes process through Gaussian Processes priors. Under appropriate smoothness and regularity assumptions on the true parameter and the nonparametric prior family, we derive explicit posterior contraction rates for the conditional intensity function and the model's parameter, in the asymptotic regime of repeatedly observed and independent sequences. Our analysis generalizes known contraction results for purely temporal Hawkes processes to the spatio-temporal setting, which allows to jointly model self-excitation across time and space in event data. These results provide, to our knowledge, the first theoretical guarantees for Bayesian nonparametric methods in spatio-temporal point data.