Real and spurious contributions for the Shannon, Rényi and Tsallis entropies

A two-parameter probability distribution is constructed by dilatation (or contraction) of the escort probability distribution. This transformation involves a physical probability distribution PP associated with the system under study and an almost arbitrary reference probability distribution P′P′. In contrast to the Shannon and Rényi entropies, the Tsallis entropy does not decompose as the sum of the physical contribution due to PP and the reference or spurious part owing to P′P′. For solving this problem, a slight modification to the relation between Tsallis and Rényi entropies must be introduced. The procedure in this paper gives rise to a nonconventional one-parameter Shannon entropy and to two-parameter Rényi and Tsallis entropies associated with PP. It also contributes to clarify the meaning and role of the escort probabilities set.