Nonparametric Bayesian estimation for multivariate Hawkes processes

Donnet, Sophie; Rivoirard, Vincent; Rousseau, Judith

Donnet, Sophie; Rivoirard, Vincent; Rousseau, Judith (2020), Nonparametric Bayesian estimation for multivariate Hawkes processes, Annals of Statistics, 48, 5, p. 2698-2727. 10.1214/19-AOS1903

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Type

Article accepté pour publication ou publié

Date

2020

Journal name

Annals of Statistics

Volume

Number

Publisher

Institute of Mathematical Statistics

Pages

2698-2727

Mathématiques et Informatique Appliquées [MIA Paris-Saclay]
Rivoirard, Vincent
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Rousseau, Judith
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]

Abstract (EN)

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. We also present a simulation study to illustrate our results and to study empirically the inference on functional connectivity graphs of neurons.

Subjects / Keywords

Multivariate counting process; Hawkes processes; nonparametric Bayesian estimation; posterior concentration rates