The Boltzmann/Shannon entropy as a measure of correlation

TL;DR

This paper shows that the entropy in statistical mechanics and information theory can be interpreted as a measure of correlation, with entropy increasing when correlations are removed between variables.

Contribution

It introduces a correlation-destroying transformation and proves that entropy does not decrease when correlations are eliminated.

Findings

Entropy is non-decreasing under correlation-destroying transformations.

The entropy of a joint distribution is less than or equal to the sum of marginal entropies.

The entropy can be viewed as a measure of correlation between variables.

Abstract

IIt is demonstrated that the entropy of statistical mechanics and of information theory, $S({\bf p}) = -\sum p_i \log p_i $ may be viewed as a measure of correlation. Given a probability distribution on two discrete variables, $p_{ij}$, we define the correlation-destroying transformation $C: p_{ij} \to \pi_{ij}$, which creates a new distribution on those same variables in which no correlation exists between the variables, i.e. $\pi_{ij} = P_i Q_j$. It is then shown that the entropy obeys the relation $S({\bf p}) \leq S({\bf \pi}) = S({\bf P}) + S({\bf Q})$, i.e. the entropy is non-decreasing under these correlation-destroying transformations.