# A Bayesian multiple changepoint model for marked poisson processes with applications to deep earthquakes

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## Abstract

A multiple changepoint model for marked Poisson process is formulated as a continuous time hidden Markov model, which is an extension of Chib’s multiple changepoint models (J Econ 86:221–241, 1998). The inference on the locations of changepoints and other model parameters is based on a two-block Gibbs sampling scheme. We suggest a continuous time version of forward-filtering backward-sampling algorithm for simulating the full trajectories of the latent Markov chain without utilizing the uniformization method. To retrieve the optimal posterior path of the latent Markov chain, i.e. the maximum a posteriori estimation of changepoint locations, a continuous-time version of Viterbi algorithm (CT-Viterbi) is proposed. The set of changepoint locations is obtainable either from the CT-Viterbi algorithm or the posterior samples of the latent Markov chain. The number of changepoints is determined according to a modified BIC criterion tailored particularly for the multiple changepoint problems of a marked Poisson process. We then perform a simulation study to demonstrate the methods. The methods are applied to investigate the temporal variabilities of seismicity rates and the magnitude-frequency distributions of medium size deep earthquakes in New Zealand.

## Keywords

Multiple changepoint models Marked temporal point process Continuous-time forward filtering backward sampling Continuous-time Viterbi algorithms Deep earthquakes b-Values## Notes

### Acknowledgements

Two referees’ suggestions are acknowledged.

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