On a simple derivation of a family of nonextensive entropies from information content

Abstract:The nonextensive entropy of Tsallis can be seen as a consequence of postulates on aself-information, i.e., the constant ratio of the first derivative of a self-information perunit probability to the curvature (second variation) of it. This constancy holds if weregard the probability distribution as the gradient of a self-information. Consideringthe form of the n th derivative of a self- information with keeping this constant ratio,we arrive at the general class of nonextensive entropies. Some properties on the seriesof entropies constructed by this picture are investigated.Keywords: Information theory; Nonadditive entropy; Information content.1 IntroductionTsallis statistics[1] can be seen as a formalism based on a pair of deformed functions of usualexponential and logarithmic ones[2, 3]. Their deformed functions are called the q -exponential andthe q -logarithmic functions, respectively. The q -exponential function is defined as e xq ≡ [1+(1 −q ) x ] 1 / 1 −q + ,where[ x ] = max{x,