May 2024 Bayesian estimation of nonlinear Hawkes processes
Déborah Sulem, Vincent Rivoirard, Judith Rousseau
Author Affiliations +
Bernoulli 30(2): 1257-1286 (May 2024). DOI: 10.3150/23-BEJ1631


Multivariate point processes (MPPs) are widely applied to model the occurrences of events, e.g., natural disasters, online message exchanges, financial transactions or neuronal spike trains. In the Hawkes process model, the probability of occurrences of future events depend on the past of the process. This model is particularly popular for modelling interactive phenomena such as disease expansion. In this work we consider the nonlinear multivariate Hawkes model, which allows to account for excitation and inhibition between interacting entities. We provide theoretical guarantees for applying nonparametric Bayesian estimation methods in this context. In particular, we obtain concentration rates of the posterior distribution on the parameters, under mild assumptions on the prior distribution and the model. These results also lead to convergence rates of Bayesian estimators. Another object of interest in event-data modelling is to infer the graph of interaction - or Granger causal graph. In this case, we provide consistency guarantees; in particular, we prove that the posterior distribution is consistent on the graph adjacency matrix of the process, as well as a Bayesian estimator based on an adequate loss function.

Funding Statement

The project leading to this work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 834175). The project is also partially funded by the EPSRC via the CDT OxWaSP.


The authors would like to thank the Editor and two anonymous referees for valuable comments and suggestions.


Download Citation

Déborah Sulem. Vincent Rivoirard. Judith Rousseau. "Bayesian estimation of nonlinear Hawkes processes." Bernoulli 30 (2) 1257 - 1286, May 2024.


Received: 1 December 2021; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699552
Digital Object Identifier: 10.3150/23-BEJ1631

Keywords: Granger-causal Graph , nonlinear Hawkes processes , nonparametric Bayesian inference


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Vol.30 • No. 2 • May 2024
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