Latent variable MCMC algorithm for the ARMA point process

Abstract

Inference for point processes when the conditional intensity depends on unobserved latent variables is generally challenging due to an intractable likelihood. An example model of this type is the autoregressive moving average (ARMA) point process, which combines self-excitement from the Hawkes process and shot-effects from the Neyman-Scott process. This process allows points of different types to influence the conditional intensity through different mechanisms. Since event types are not part of the observed data, a Bayesian treatment of model inference is proposed that includes latent variables via data augmentation. The latent variables represent the genealogical tree that connects the points, as immigrants or offspring, due to the model's connection to a branching process. We develop purpose-built Hamiltonian Monte Carlo algorithms for the parameter updates within a Gibbs sampler. An application to a Japan earthquake catalogue is presented, highlighting the versatility of the model.