Efficient Non-parametric Bayesian Hawkes Processes

In this paper, we develop an efficient nonparametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their uncertainty. Our method is based on the cluster representation of Hawkes processes. Utilizing the stationarity of the Hawkes process, we efficiently sample random branching structures and thus, we split the Hawkes process into clusters of Poisson processes. We derive two algorithms -- a block Gibbs sampler and a maximum a posteriori estimator based on expectation maximization -- and we show that our methods have a linear time complexity, both theoretically and empirically. On synthetic data, we show our methods to be able to infer flexible Hawkes triggering kernels. On two large-scale Twitter diffusion datasets, we show that our methods outperform the current state-of-the-art in goodness-of-fit and that the time complexity is linear in the size of the dataset. We also observe that on diffusions related to online videos, the learned kernels reflect the perceived longevity for different content types such as music or pets videos

Efficient non-parametric Bayesian hawkes processes

© 2019 International Joint Conferences on Artificial Intelligence. All rights reserved. In this paper, we develop an efficient nonparametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their uncertainty. Our method is based on the cluster representation of Hawkes processes. Utilizing the stationarity of the Hawkes process, we efficiently sample random branching structures and thus, we split the Hawkes process into clusters of Poisson processes. We derive two algorithms - a block Gibbs sampler and a maximum a posteriori estimator based on expectation maximization - and we show that our methods have a linear time complexity, both theoretically and empirically. On synthetic data, we show our methods to be able to infer flexible Hawkes triggering kernels. On two large-scale Twitter diffusion datasets, we show that our methods outperform the current state-of-the-art in goodness-of-fit and that the time complexity is linear in the size of the dataset. We also observe that on diffusions related to online videos, the learned kernels reflect the perceived longevity for different content types such as music or pets videos

Approximate Inference for Non-parametric Bayesian Hawkes Processes and Beyond

The Hawkes process has been widely applied to modeling self-exciting events including neuron spikes, earthquakes and tweets. To avoid designing parametric triggering kernels, the non-parametric Hawkes process has been proposed, in which the triggering kernel is in a non-parametric form. However, inference in such models suffers from poor scalability to large-scale datasets and sensitivity to uncertainty in the random finite samples. To deal with these issues, we employ Bayesian non-parametric Hawkes processes and propose two kinds of efficient approximate inference methods based on existing inference techniques. Although having worked as the cornerstone of probabilistic methods based on Gaussian process priors, most of existing inference techniques approximately optimize standard divergence measures such as the Kullback-Leibler (KL) divergence, which lacks the basic desiderata for the task at hand, while chiefly offering merely technical convenience. In order to improve them, we further propose a more advanced Bayesian inference approach based on the Wasserstein distance, which is applicable to a wide range of models. Apart from these works, we also explore a robust frequentist estimation method beyond the Bayesian field. Efficient inference techniques for the Hawkes process will help all the different applications that it already has, from earthquake forecasting, finance to social media. Furthermore, approximate inference techniques proposed in this thesis have the potential to be applied to other models to improve robustness and account for uncertainty