Using Bayesian belief networks to analyse the stochastic dependence between interevent time and size of earthquakes

The purpose of this article is to show how Bayesian belief networks can be used in analysis of the sequence of the earthquakes which have occurred in a region, to study the interaction among the variables characterizing each event. These relationships can be represented by means of graphs consisting of vertices and edges; the vertices correspond to random variables, while the edges express properties of conditional independence. We have examined Italian seismicity as reported in two data bases, the NT4.1.1 catalogue and the ZS.4 zonation, and taken into account three variables: the size of the quake, the time elapsed since the previous event, and the time before the subsequent one. Assigning different independence relationships among these variables, first two couples of bivariate models, and then eight trivariate models have been defined. After presenting the main elements constituting a Bayesian belief network, we introduce the principal methodological aspects concerning estimation and model comparison. Following a fully Bayesian approach, prior distributions are assigned on both parameters and structures by combining domain knowledge and available information on homogeneous seismogenic zones. Two case studies are used to illustrate in detail the procedure followed to evaluate the fitting of each model to the data sets and compare the performance of alternative models. All eighty Italian seismogenic zones have been analysed in the same way; the results obtained are reported briefly. We also show how to account for model uncertainty in predicting a quantity of interest, such as the time of the next event.