Nonparametric Bayesian estimation for multivariate Hawkes processes

Sophie Donnet (1) , Vincent Rivoirard (2) , Judith Rousseau (2)

1 MIA Paris-Saclay - Mathématiques et Informatique Appliquées
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 L-1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L-1-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.

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

Domaines: Statistiques [math.ST]

Sophie Donnet, Vincent Rivoirard, Judith Rousseau. Nonparametric Bayesian estimation for multivariate Hawkes processes. Annals of Statistics, 2020, 48 (5), pp.2698-2727. ⟨hal-03320970⟩