# Generalized Mutual Information

## TL;DR

This paper introduces a family of generalized mutual information measures that are finitely defined for all distributions on countable alphabets, addressing a fundamental limitation in classical mutual information.

## Contribution

It proposes a new family of mutual information measures that are universally finite and retain key properties, filling a foundational gap in information theory.

## Key findings

- All members are finitely defined for all distributions except Shannon's mutual information.
- Retain utility and properties of finite Shannon mutual information.
- Address a fundamental void in the theoretical foundation of information theory.

## Abstract

Mutual information is one of the essential building blocks of information theory. Yet, it is only finitely defined for distributions with fast decaying tails on a countable joint alphabet of two random elements. The unboundedness of mutual information over the general class of all distributions on a joint alphabet prevents its potential utility to be fully realized. This is in fact a void in the foundation of information theory that needs to be filled. This article proposes a family of generalized mutual information all of whose members 1) are finitely defined for each and every distribution of two random elements on a joint countable alphabet, except the one by Shannon, and 2) enjoy all utilities of a finite Shannon's mutual information.