A Divergence Formula for Randomness and Dimension (Short Version)

TL;DR

This paper introduces a divergence formula linking the dimension of random sequences to the Shannon entropy and Kullback-Leibler divergence between probability measures, providing a new way to measure measure similarity.

Contribution

It establishes a divergence formula connecting the dimension of sequences, randomness, and measure similarity using Shannon entropy and Kullback-Leibler divergence.

Findings

Proves the divergence formula for computable, positive probability measures.

Shows the relation between dimension, entropy, and divergence for random sequences.

Provides a new measure of similarity between probability measures.

Abstract

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $\beta$ is the uniform probability measure. This paper shows that $\dim^\beta(S)$ and its dual $\Dim^\beta(S)$, the {\it strong dimension} of $S$ with respect to $\beta$, can be used in conjunction with randomness to measure the similarity of two probability measures $\alpha$ and $\beta$ on $\Sigma$. Specifically, we prove that the {\it divergence formula} $$\dim^\beta(R) = \Dim^\beta(R) =\CH(\alpha) / (\CH(\alpha) + \D(\alpha || \beta))$$ holds whenever $\alpha$ and $\beta$ are computable, positive probability measures on $\Sigma$ and $R \in \Sigma^\infty$ is random with respect to $\alpha$. In this formula, $\CH(\alpha)$ is the Shannon entropy of $\alpha$, and $\D(\alpha||\beta)$ is the Kullback-Leibler divergence between $\alpha$ and $\beta$.