Nonparametric Bayesian estimation of multivariate Hawkes processes

TL;DR

This paper develops Bayesian nonparametric methods for estimating multivariate Hawkes processes, deriving posterior concentration rates and illustrating applications in neural connectivity inference.

Contribution

It introduces new Bayesian nonparametric estimation techniques for multivariate Hawkes processes with theoretical convergence rates and practical neural connectivity applications.

Findings

Derived posterior concentration rates for stochastic intensities.

Established convergence rates for interaction functions.

Demonstrated numerical applications in neural connectivity inference.

Abstract

This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.