  The Hawkes process (HP) has been widely applied to modeling self-exciting
events including neuron spikes, earthquakes and tweets. To avoid designing
parametric triggering kernel and to be able to quantify the prediction
confidence, the non-parametric Bayesian HP has been proposed. However, the
inference of such models suffers from unscalability or slow convergence. In
this paper, we aim to solve both problems. Specifically, first, we propose a
new non-parametric Bayesian HP in which the triggering kernel is modeled as a
squared sparse Gaussian process. Then, we propose a novel variational inference
schema for model optimization. We employ the branching structure of the HP so
that maximization of evidence lower bound (ELBO) is tractable by the
expectation-maximization algorithm. We propose a tighter ELBO which improves
the fitting performance. Further, we accelerate the novel variational inference
schema to linear time complexity by leveraging the stationarity of the
triggering kernel. Different from prior acceleration methods, ours enjoys
higher efficiency. Finally, we exploit synthetic data and two large social
media datasets to evaluate our method. We show that our approach outperforms
state-of-the-art non-parametric frequentist and Bayesian methods. We validate
the efficiency of our accelerated variational inference schema and practical
utility of our tighter ELBO for model selection. We observe that the tighter
ELBO exceeds the common one in model selection.
