On the principle of maximum entropy and the earthquake frequency–magnitude relation

Summary. The entropy S for a continuous distribution p ( x ) is defined by This expression, however, is a measure of uncertainty relative to the coordinate x so that the probability distribution p ( x ) generated from the principle of maximum entropy depends on the choice of x . Only when the chosen parameter actually has a uniform prior distribution, can we expect the generated distribution to conform with the empirical data. For a physical system in which the independent variable x is measured to only limited accuracy, the prior distribution m ( x ) can be shown to be inversely proportional to the measurement error of x . A parameter with uniform prior distribution, then, is one that can be measured with equal accuracy throughout its range. In this context, the magnitude of an earthquake is such a parameter because using this parameter in the principle of maximum entropy leads to the empirically determined Gutenberg‐Richter frequency‐magnitude relation. Other proposed frequency‐magnitude relations can also be generated from the principle of maximum entropy by imposing appropriate constraints. However, it is emphasized that such relations are generated from the principle as null hypotheses, to be tested by the empirical regional or global seismicity data.