Earthquakes are regarded as the realization of a point process modeled by a
generalized Poisson distribution. We assume that the Gutenberg-Richter law
describes the magnitude distribution of all the earthquakes in a sample, w
ith a constant b value. We model the occurrence rate density of earthquakes
in space and time as the sum of two terms, one representing the independen
t, or spontaneous, activity and the other representing the activity induced
by previous earthquakes. The first term depends only on space and is model
ed by a continuous function of the geometrical coordinates, obtained by smo
othing the discrete distribution of the past instrumental seismicity. The s
econd term also depends on time, and it is factorized in two terms that dep
end on the space distance (according to an isotropic normal distribution) a
nd on the time difference (according to the generalized Omori law), respect
ively, from the past earthquakes. Knowing the expected rate density, the li
kelihood of any realization of the process (actually represented by an eart
hquake catalog) can be computed straightforwardly. This algorithm was used
in two ways: (1) during the learning phase, for the maximum likelihood esti
mate of the few free parameters of the model, and (2) for hypothesis testin
g. For the learning phase we used the catalog of Italian seismicity (M grea
ter than or equal to3.5) from May 1976 to December 1998. The model was test
ed on a new and independent data set (January-December 1999). We demonstrat
ed for this short time period that in the Italian region this time-dependen
t model has a significantly better performance than a stationary Poisson mo
del, even if its likelihood is computed excluding the obvious component of
main shock-aftershock interaction.