Multivariate point processes are widely applied to model event-type data such as natural disasters, online message exchanges, financial transactions or neuronal spike trains. One very popular point process model in which the probability of occurrences of new events depend on the past of the process is the Hawkes process. In this work we consider the nonlinear Hawkes process, which notably models excitation and inhibition phenomena between dimensions of the process. In a nonparametric Bayesian estimation framework, 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 recover the graph of interaction - or Granger connectivity graph - of the phenomenon. We provide consistency guarantees on Bayesian methods for estimating this quantity; 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.

翻译：多元点过程被广泛应用于建模事件型数据，例如自然灾害、在线信息交换、金融交易或神经元脉冲序列。霍克斯过程作为一种流行的点过程模型，其中新事件发生的概率依赖于过程的历史。本文考虑非线性霍克斯过程，该过程特别能够建模过程维度之间的激发与抑制现象。在非参数贝叶斯估计框架下，我们在先验分布和模型的温和假设条件下，获得了参数后验分布的收敛速度。这些结果也导出了贝叶斯估计量的收敛速度。事件数据建模的另一个重要目标是恢复现象的交互图（即格兰杰因果连通图）。我们给出了贝叶斯方法用于估计该量的相合性保证；特别地，我们证明了过程邻接矩阵的后验分布具有相合性，并证明了基于适当损失函数的贝叶斯估计量也具有相合性。