Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.

翻译：霍克斯过程常被用于建模多元事件数据（如神经元脉冲序列、社交互动和金融交易）中的依赖与交互现象。在非参数设定下，学习霍克斯过程的时间依赖结构通常是一项计算成本高昂的任务，尤其是在贝叶斯估计方法中。对于广义非线性霍克斯过程而言，应用马尔可夫链蒙特卡洛方法计算双重难解的后验分布，在实践中难以扩展至高维过程。近期，已有研究者提出针对后验分布的均值场变分近似的高效算法。本文首先将现有变分贝叶斯方法统一于一个通用的非参数推断框架下，并在先验、变分类别及非线性模型的可验证条件下分析这些方法的渐近性质。其次，我们提出一种新的稀疏诱导过程，并为流行的Sigmoid霍克斯过程推导出自适应均值场变分算法。该算法具有可并行化特性，因此在高维场景下计算高效。通过大量数值模拟实验，我们进一步证明该方法能够自适应于霍克斯过程参数的维度，并对特定类型的模型误设具有部分鲁棒性。