Multivariate Hawkes Processes (MHPs) are a class of point processes that can
account for complex temporal dynamics among event sequences. In this work, we
study the accuracy and computational efficiency of three classes of algorithms
which, while widely used in the context of Bayesian inference, have rarely been
applied in the context of MHPs: stochastic gradient expectation-maximization,
stochastic gradient variational inference and stochastic gradient Langevin
Monte Carlo. An important contribution of this paper is a novel approximation
to the likelihood function that allows us to retain the computational
advantages associated with conjugate settings while reducing approximation
errors associated with the boundary effects. The comparisons are based on
various simulated scenarios as well as an application to the study the risk
dynamics in the Standard & Poor's 500 intraday index prices among its 11
sectors