Additive Entropies of degree-q and the Tsallis Entropy

The Tsallis entropy is shown to be an additive entropy of degree-q that information scientists have been using for almost forty years. Neither is it a unique solution to the nonadditive functional equation from which random entropies are derived. Notions of additivity, extensivity and homogeneity are clarified. The relation between mean code lengths in coding theory and various expressions for average entropies is discussed. 1 The ‘Tsallis’ Entropy In 1988 Tsallis [1]published a much quoted paper containing an expression for the entropy which differed from the usual one used in statistical mechanics. Previous to this, the Rényi entropy was used as an interpolation formula that connected the Hartley–Boltzmann entropy to the Shannon–Gibbs entropy. Notwithstanding the fact that the Rényi entropy is additive, it lacks many other properties that characterize the Shannon–Gibbs entropy. For example, the Rényi entropy is not subadditive, recursive, nor does it possess the branching and sum properties [2]. The so-called Tsallis entropy fills this gap, while being nonadditive, it has many other properties that resemble the Shannon– Gibbs entropy. It is no wonder then that this entropy fills an important gap. Yet, it appears odd, to say the least, that information scientists have left such a gaping void in their analysis of entropy functions. A closer analysis of the literature reveals that this is not the case and, indeed, a normalized Tsallis entropy seems to have first appeared in a 1967 paper by Havrda and Charvát [3] who introduced the normalized ‘Tsallis’ entropy Sn,q(p1, . . . , pn) = ( n