Skip to content Skip to navigation

OpenStax-CNX

You are here: Home » Content » Mutual Information

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Digital Communication Systems"

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Recently Viewed

This feature requires Javascript to be enabled.
 

Mutual Information

Module by: Behnaam Aazhang. E-mail the author

Summary: A description of mutual information between two random variables with examples.

Recall that

HXY= x x y yp X Y xylogp X Y xy H X Y x x y y p X Y x y p X Y x y
(1)
HY+H X | Y =HX+H Y | X H Y H X | Y H X H Y | X
(2)

Definition 1: Mutual Information
The mutual information between two discrete random variables is denoted by X;Y X Y and defined as
X;Y=HXH X | Y X Y H X H X | Y
(3)
Mutual information is a useful concept to measure the amount of information shared between input and output of noisy channels.

In our previous discussions it became clear that when the channel is noisy there may not be reliable communications. Therefore, the limiting factor could very well be reliability when one considers noisy channels. Claude E. Shannon in 1948 changed this paradigm and stated a theorem that presents the rate (speed of communication) as the limiting factor as opposed to reliability.

Example 1

Consider a discrete memoryless channel with four possible inputs and outputs.

Figure 1
Figure 1 (Figure7-32.png)

Every time the channel is used, one of the four symbols will be transmitted. Therefore, 2 bits are sent per channel use. The system, however, is very unreliable. For example, if "a" is received, the receiver can not determine, reliably, if "a" was transmitted or "d". However, if the transmitter and receiver agree to only use symbols "a" and "c" and never use "b" and "d", then the transmission will always be reliable, but 1 bit is sent per channel use. Therefore, the rate of transmission was the limiting factor and not reliability.

This is the essence of Shannon's noisy channel coding theorem, i.e., using only those inputs whose corresponding outputs are disjoint (e.g., far apart). The concept is appealing, but does not seem possible with binary channels since the input is either zero or one. It may work if one considers a vector of binary inputs referred to as the extension channel.

X input vector= X 1 X 2 X n X ¯ n=01n X input vector X 1 X 2 X n X ¯ n 0 1 n

Y output vector= Y 1 Y 2 Y n Y ¯ n=01n Y output vector Y 1 Y 2 Y n Y ¯ n 0 1 n

Figure 2
Figure 2 (Figure7-34.png)

This module provides a description of the basic information necessary to understand Shannon's Noisy Channel Coding Theorem. However, for additional information on typical sequences, please refer to Typical Sequences.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks