# On extractable shared information

## TL;DR

This paper introduces a new measure called extractable shared information that is left monotonic, providing a novel way to decompose mutual information into shared, complementary, and unique parts, with implications for information theory.

## Contribution

It proposes a new left monotonic shared information measure and a corresponding nonnegative mutual information decomposition, advancing understanding of information sharing.

## Key findings

- The measure is left monotonic and bounds shared information about $S$ by that about $f(S)$.
- The decomposition separates mutual information into shared, complementary, and unique components.
- Left monotonic shared information conflicts with the Blackwell interpretation of unique information.

## Abstract

We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information, that is left monotonic; that is, the information shared about $S$ is bounded from below by the information shared about $f(S)$ for any function $f$. We show that our measure leads to a new nonnegative decomposition of the mutual information $I(S;X_1X_2)$ into shared, complementary and unique components. We study properties of this decomposition and show that a left monotonic shared information is not compatible with a Blackwell interpretation of unique information. We also discuss whether it is possible to have a decomposition in which both shared and unique information are left monotonic.