Mean-field inference of Hawkes point processes

TL;DR

This paper introduces a fast mean-field based estimation method for Hawkes point processes that is accurate under certain conditions, offering a computationally efficient alternative to traditional maximum likelihood estimation.

Contribution

The authors develop a novel mean-field approximation approach for parameter estimation in high-dimensional Hawkes processes, significantly improving computational speed while maintaining accuracy.

Findings

Estimator is biased but comparable in precision to MLE

Method is faster and effective in high-dimensional or weak interaction regimes

Theoretical analysis confirms accuracy and efficiency of the approach

Abstract

We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d-dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the Maximum Likelihood Estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters.