Bayesian estimation for a parametric Markov renewal model applied to seismic data

This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its implementation. The inter-occurrence times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on "learning data" and moment existence conditions. This framework is applied to data of earthquakes of three types of severity (low, medium and high size) that occurred in the central Northern Apennines in Italy and collected by the CPTI04 (2004) catalogue. Assumptions on two types of energy accumulation and release mechanisms are evaluated.