Estimation in linear high dimensional Hawkes processes: a Bayesian approach

TL;DR

This work develops a Bayesian framework for estimating high-dimensional linear Hawkes processes under sparsity, delivering the first contraction results for the $L_1$-norm on the parameters and for the empirical $L_1$ loss on intensities. By using product-form priors with model-selection structures over active edges, the authors enable scalable posterior computation and derive rates that depend on sparsity $s_0$, graph size $K$, and sparsity-inducing quantities $L_T$. They establish two main contraction results: one for the empirical $L_1$ distance $d_{1,T}$ and another for the direct $L_1$ distance on the parameter vector $f$, with different regimes for $oldsymbol{\epsilon_T}$ and the log-dimension term $ ext{log }K$. A two-step procedure further improves rates when $K$ is very large, exploiting posterior information about edge masses to refine the active graph; nonparametric priors on interaction functions, such as random-histogram or spline-based models, yield concrete rates tied to smoothness. Overall, the paper provides frequentist guarantees for Bayesian procedures in high-dimensional Hawkes models, offering practical prior constructions and scalable inference strategies relevant to neuroscience and finance applications.

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.

Theorems & Definitions (19)

• Lemma 2.4
• Proof 1: Proof of Lemma \ref{['lem:lambda2']}
• Definition 2.5: Model selection prior
• Theorem 2.6
• Proof 2: Proof of Theorem \ref{['th:gene:d1T']}
• Remark 2.7
• Remark 2.8
• Theorem 2.10
• Remark 2.11
• Proof 3: Proof of Theorem \ref{['th:gene:L1']}