Significance estimation for the Kullback-Leibler divergence: the Poissonian case in seismological studies

The Kullback-Leibler divergence, κ, is a widely used measure of the difference between an observed probability distribution and a reference one; κ=0 when the two distributions are equal, but it has no upper limit to help interpret the significance of any other κ value. Using as an example the problem of distinguishing clustering or gaps in the time occurrence of earthquakes from seismicity uniformly distributed in time, a Monte Carlo method for evaluating the significance of a particular κ value is presented, a method that takes into account the number of classes in the distributions and the length of the sample. Application of this method yields a probability according to which the hypothesis of the observed distribution being a realization of the reference one can be discarded or accepted with a quantitative degree of confidence. This method, and two possible reference values, are presented using the Poisson distribution as an example, but they can be used for other reference distributions.