Efficient EM-variational inference for nonparametric Hawkes process

The classic Hawkes process assumes the baseline intensity to be constant and the triggering kernel to be a parametric function. Differently, we present a generalization of the parametric Hawkes process by using a Bayesian nonparametric model called quadratic Gaussian Hawkes process . We model the baseline intensity and trigger kernel as the quadratic transformation of random trajectories drawn from a Gaussian process (GP) prior. We derive an analytical expression for the EM-variational inference algorithm by augmenting the latent branching structure of the Hawkes process to embed the variational Gaussian approximation into the EM framework naturally. We also use a series of schemes based on the sparse GP approximation to accelerate the inference algorithm. The results of synthetic and real data experiments show that the underlying baseline intensity and triggering kernel can be recovered efficiently and our model achieved superior performance in fitting capability and prediction accuracy compared to the state-of-the-art approaches.