Information and coding discrimination of pseudo-additive entropies (PAE)

TL;DR

Pseudo-additive entropies (PAE) are limited in their applicability for generalized statistical mechanics and information theory, especially regarding independence and code length bounds, compared to Rényi entropy.

Contribution

The paper clarifies the limitations of PAE in statistical mechanics and information theory, contrasting their bounds and applicability with Rényi entropy.

Findings

PAE cannot serve as a basis for generalized statistical mechanics.

PAE's relation to Rényi entropy is unbounded as code length increases.

PAE has a more limited validity range than Rényi entropy.

Abstract

PAE cannot be made a basis for either a generalized statistical mechanics or a generalized information theory. Either statistical independence must be waived, or the expression of the averaged conditional probability as the difference between the marginal and joint entropies must be relinquished. The same inequality, relating the PAE to the R\'enyi entropy, when applied to the mean code length produces an expression that it is without bound as the order of the code length approaches infinity. Since the mean code length associated with the R\'enyi entropy is finite and can be made to come as close to the Hartley entropy as desired in the same limit, the PAE have a more limited range of validity than the R\'enyi entropy which they approximate.