Bayesian inference of a doubly stochastic Poisson process with a non-stationary state process

A doubly stochastic Poisson process is proposed in order to study seismic sequences. It is a state-space model where the state process jumps among three possible states, each of them characterised by a different point process constituting the observed process. Hence we observe series of subsequent realizations of different point processes and we aim to estimate which state is active at any time t. To better model the real phenomenon we decided to enrich the models previously analysed by considering a non-stationary state process. The non-stationarity arises from the assumption that the current state depends on both the past observations and the time of the last state jump. A sequential Monte Carlo method is applied to approximate the likelihood function and Markov Chain Monte Carlo methods are used for the parameter estimation.