Estimation in linear high dimensional Hawkes processes: a Bayesian approach

TL;DR

This paper investigates Bayesian methods for estimating high-dimensional linear Hawkes processes, providing the first theoretical results on parameter recovery in such settings, especially under sparsity.

Contribution

It introduces novel theoretical analysis of Bayesian procedures for high-dimensional Hawkes processes, focusing on $L_1$ norm control of parameters in sparse regimes.

Findings

First to control $L_1$ norm on parameters in this framework

First to analyze Bayesian procedures in high-dimensional Hawkes processes

Establishes theoretical properties of Bayesian estimation in sparse settings

Abstract

In this paper we study the frequentist properties of Bayesian approaches in linear high dimensional Hawkes processes in a sparse regime where the number of interaction functions acting on each component of the Hawkes process is much smaller than the dimension. We consider two types of loss function: the empirical $L_1$ distance between the intensity functions of the process and the $L_1$ norm on the parameters (background rates and interaction functions). Our results are the first results to control the $L_1$ norm on the parameters under such a framework. They are also the first results to study Bayesian procedures in high dimensional Hawkes processes.