Common information revisited

TL;DR

This paper explores the nuanced relationship between mutual and common information, providing quantitative bounds and constructions that deepen understanding of information sharing and reconstruction in probabilistic models.

Contribution

It establishes a quantitative version of the Gács–Körner theorem using hypercontractivity, analyzes tradeoffs in information reconstruction, and constructs worst-case distributions.

Findings

Quantitative bounds on mutual and common information relationships

Tradeoff analysis for information reconstruction with Hamming distance constraints

Construction of worst-case distributions for information tradeoffs

Abstract

One of the main notions of information theory is the notion of mutual information in two messages (two random variables in Shannon information theory or two binary strings in algorithmic information theory). The mutual information in $x$ and $y$ measures how much the transmission of $x$ can be simplified if both the sender and the recipient know $y$ in advance. G\'acs and K\"orner gave an example where mutual information cannot be presented as common information (a third message easily extractable from both $x$ and $y$). Then this question was studied in the framework of algorithmic information theory by An. Muchnik and A. Romashchenko who found many other examples of this type. K. Makarychev and Yu. Makarychev found a new proof of G\'acs--K\"orner results by means of conditionally independent random variables. The question about the difference between mutual and common information can be studied quantitatively: for a given $x$ and $y$ we look for three messages $a$, $b$, $c$ such that $a$ and $c$ are enough to reconstruct $x$, while $b$ and $c$ are enough to reconstruct $y$. In this paper: We state and prove (using hypercontractivity of product spaces) a quantitative version of G\'acs--K\"orner theorem; We study the tradeoff between $\abs{a}, \abs{b}, \abs{c}$ for a random pair $(x, y)$ such that Hamming distance between $x$ and $y$ is $\eps n$ (our bounds are almost tight); We construct "the worst possible" distribution on $(x, y)$ in terms of the tradeoff between $\abs{a}, \abs{b}, \abs{c}$.