# A Tight Upper Bound on Mutual Information

## TL;DR

This paper presents a mathematically tight upper bound on mutual information between signals and channel outputs, based on the joint distribution and MAP decoding, aiding in data analysis where mutual information is estimated.

## Contribution

It introduces a new tight upper bound on mutual information derived from the joint distribution and MAP decoding, advancing theoretical understanding.

## Key findings

- Derived a tight upper bound on mutual information.
- Characterized properties of distributions that maximize mutual information.
- Provides a theoretical tool for data analysis involving mutual information.

## Abstract

We derive a tight lower bound on equivocation (conditional entropy), or equivalently a tight upper bound on mutual information between a signal variable and channel outputs. The bound is in terms of the joint distribution of the signals and maximum a posteriori decodes (most probable signals given channel output). As part of our derivation, we describe the key properties of the distribution of signals, channel outputs and decodes, that minimizes equivocation and maximizes mutual information. This work addresses a problem in data analysis, where mutual information between signals and decodes is sometimes used to lower bound the mutual information between signals and channel outputs. Our result provides a corresponding upper bound.