Parametric inference for the discretely observed multivariate Hawkes process using particle Markov Chain Monte Carlo

The multivariate Hawkes process (MHP) is a useful statistical model for analysing multidimensional event time sequences that exhibit self-excitation and cross-excitation. When the MHP is monitored discretely, only the total number of events for each dimension in disjoint time intervals is observed. The likelihood function relative to this data is intractable, so traditional inference techniques are not available. To address this, we design an unbiased estimate of the intractable likelihood function using sequential Monte Carlo (SMC) based on a representation of the unobserved event times as latent variables in a state-space model. The unbiasedness of the SMC estimate allows for its use in place of the true likelihood in a Metropolis-Hastings algorithm, enabling the construction of a Markov Chain Monte Carlo sample from the posterior distribution over the parameters of the MHP. Using simulated data, we assess the performance of our method and demonstrate that it outperforms existing approaches in terms of mean squared error and computational efficiency. Terrorist activity in Afghanistan and Pakistan from 2018 to 2021 is analysed based on daily count data to examine the dynamics of terrorism in the region.