Qualms concerning Tsallis’ Use of the Maximum Entropy Formalism

Tsallis’ ‘statistical thermodynamic’ formulation of the nonadditive entropy of degree-α is neither correct nor self-consistent. It is well known that the maximum entropy formalism [1], the minimum discrimination information [2], and Gauss’ principle [3, 4] all lead to the same results when a certain condition on the prior probability distribution is imposed [5]. All these methods lead to the same form of the posterior probability distribution; namely, the exponential family of distributions. Tsallis and collaborators [6] have tried to adapt the maximum entropy formalism that uses the Shannon entropy to one that uses a nonadditive entropy of degree-α. In order to come out with analytic expressions for the probabilities that maximize the nonadditive entropy they found it necessary to use ‘escort probabilities’[7] of the same power as the nonadditive entropy. If the procedure they use is correct then it follows that Gauss’ principle should give the same optimum probabilities. Yet, we will find that the Tsallis result requires that the prior probability distribution be given by the same unphysical condition as the maximum entropy formalism and, what is worse, the potential of the error law be required to vanish. The potential of the error law is what information theory refers to as the error [8]; that is, the difference between the inaccuracy and the entropy. Unless the ‘true’ probability distribution, P = (p(x1), p(x2) . . . , p(xm)) coincides with the estimated probability distribution, Q = (q(x1), q(x2), . . . q(xm)), the error does not vanish. Moreover, we shall show that two procedures of averaging, one using the escort probabilities explicitly, do not give the same result, and the relation between the potential of the error law and the nonadditive entropy requires the latter to vanish when the former vanishes. Let X be a random variable whose values x1, x2, . . . , xm are obtained at m independent trials. Prior to the observations the distribution is Q, and after the observations the unknown probability distribution is P . The observer has