Scalable and adaptive variational Bayes methods for Hawkes processes

TL;DR

This paper introduces scalable, adaptive variational Bayes methods for nonlinear Hawkes processes, unifying existing approaches, analyzing their properties, and proposing a new sparsity-inducing algorithm suitable for high-dimensional data.

Contribution

It unifies variational Bayes methods under a general framework, analyzes their asymptotic properties, and proposes a novel adaptive, parallelisable algorithm for sigmoid Hawkes processes.

Findings

Algorithm is computationally efficient and parallelisable.

Method adapts to the dimensionality of the data.

Procedure shows robustness to some model mis-specification.

Abstract

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.