Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy

TL;DR

This paper establishes the combinatorial basis as the fundamental foundation of entropy and cross-entropy, deriving classical functions and proposing generalized definitions for broader applications in probabilistic systems.

Contribution

It introduces a combinatorial basis that generalizes entropy and cross-entropy, enabling analysis of diverse probabilistic systems beyond multinomial models.

Findings

Derivation of Shannon entropy and Kullback-Leibler divergence from multinomial distributions.

Introduction of generalized entropy and cross-entropy definitions.

Re-examination of Jaynes' statistical mechanics formulation using combinatorial principles.

Abstract

This study critically analyses the information-theoretic, axiomatic and combinatorial philosophical bases of the entropy and cross-entropy concepts. The combinatorial basis is shown to be the most fundamental (most primitive) of these three bases, since it gives (i) a derivation for the Kullback-Leibler cross-entropy and Shannon entropy functions, as simplified forms of the multinomial distribution subject to the Stirling approximation; (ii) an explanation for the need to maximize entropy (or minimize cross-entropy) to find the most probable realization; and (iii) new, generalized definitions of entropy and cross-entropy - supersets of the Boltzmann principle - applicable to non-multinomial systems. The combinatorial basis is therefore of much broader scope, with far greater power of application, than the information-theoretic and axiomatic bases. The generalized definitions underpin a new discipline of ``{\it combinatorial information theory}'', for the analysis of probabilistic systems of any type. Jaynes' generic formulation of statistical mechanics for multinomial systems is re-examined in light of the combinatorial approach. (abbreviated abstract)