In this paper we study the frequentist properties of Bayesian approaches in linear high dimensional Hawkes processes in a sparse regime where the number of interaction functions acting on each component of the Hawkes process is much smaller than the dimension. We consider two types of loss function: the empirical $L_1$ distance between the intensity functions of the process and the $L_1$ norm on the parameters (background rates and interaction functions). Our results are the first results to control the $L_1$ norm on the parameters under such a framework. They are also the first results to study Bayesian procedures in high dimensional Hawkes processes.

翻译：本文研究了稀疏机制下线性高维霍克斯过程中贝叶斯方法的频率性质，其中作用于霍克斯过程各分量的交互函数数量远小于维度。我们考虑两类损失函数：过程强度函数的经验$L_1$距离，以及参数（背景速率与交互函数）的$L_1$范数。我们的成果首次在此框架下实现了对参数$L_1$范数的控制，同时也是首次针对高维霍克斯过程的贝叶斯方法研究。