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Many complex phenomena concerning extreme events causing natural disasters exhibit power-law behavior, reflecting a hierarchical or fractal structure. These phenomena seem to be susceptible to description using approaches inspired by statistical mechanics, particularly approaches involving generalizations of the classical concept of entropy. In this framework we consider the nonextensive Tsallis entropy; by its constrained optimization we derive the expression of the q- exponential distribution which controls the behavior of variables that, in geophysics, can be for instance seismic moment, length of faults, and inter-event times. We estimate the model parameters in the Bayesian perspective and apply this approach to the identification of different dynamical regimes in seismic sequences.

The historical seismic data are suitably modelled by the self-correcting point processes whose conditional intensity functions give the instantaneous occurrence probability of at least one event. These models assume that the level of some physical quantity in a region is revealing of the proneness to generate earthquakes in the immediate future. Four versions of this self-correcting point process are defined through four different proposals for the quantity accumulated in the level of the process. Bayesian inference is performed.

Despite the large number of studies on the time-dependent hazard a commonly accepted model does not exist yet, but there are some more shared conjectures as the dependence on the time elapsed since the last occurrence, that is, the idea that after each event the probability of the future process may be the same. From the probabilistic point of view this means to consider renewal processes. They are appropriate to model large earthquakes after which one can assume that stress accumulation process restarts. Statistically this implies that the times between large seismic events can be considered as realizations of independent, identically distributed random variables, the interoccurrence times. Various probability distributions have been considered in the literature for these times. When satisfactory results in terms of fitting were obtained, then one looked for physical interpretations of the underlying process; for instance, distributions with decreasing hazard can support the idea that there are strong interactions among neighbouring fault segments. We prefer the opposite way of dealing with the problem: we do not make any assumption on the functional form of the distribution of the interoccurrence time and strengthen the probabilistic methodology by considering such a distribution as a random function modelled by a Polya process. The approach followed for estimation is Bayesian, semi-parametric; we just assume that the prior expectation of the unknown density function is chosen in the class of the generalized gamma distributions for its extremely great flexibility. Renewal models have to be applied in intraplate regions for global hazard assessment and for decade-long forecasts.

In this paper we address some issues arising in the implementation of Markov chain Monte Carlo methods; in particular we analyse whether the choice of transition kernels depending on a specific problem speeds up the convergence of a Metropolis-Hastings-type algorithm. This approach is applied to the retrospective detection of multiple structural changes in the physical process generating earthquakes. As the number of changes is unknown, the adopted hierarchical Bayesian model has variable-dimension parameters. The sensitivity of the method and issues related to the estimation of both the parameters and the posterior model distributions are also dealt with.