Computing and Communications 2. Information Theory -Entropy

1896 1920 1987 2006

Computing and Communications2. Information Theory

-Entropy

Ying Cui

Department of Electronic Engineering

Shanghai Jiao Tong University, China

2017, Autumn

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Outline

• Entropy

• Joint entropy and conditional entropy

• Relative entropy and mutual information

• Relationship between entropy and mutual information

• Chain rules for entropy, relative entropy and mutual information

• Jensen’s inequality and its consequences

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Reference

• Elements of information theory, T. M. Cover and J. A. Thomas, Wiley

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OVERVIEW

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Information Theory

• Information theory answers two fundamental questions in communication theory

– what is the ultimate data compression?

-- entropy H

– what is the ultimate transmission rate of communication? -- channel capacity C

• Information theory is considered as a subset of communication theory

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Information Theory

• Information theory has fundamental contributions to other fields

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A Mathematical Theory of Commun.

• In 1948, Shannon published “A Mathematical Theory of Communication”, founding Information Theory

• Shannon made two major modifications having huge impact on communication design

– the source and channel are modeled probabilistically

– bits became the common currency of communication

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A Mathematical Theory of Commun.

• Shannon proved the following three theorems– Theorem 1. Minimum compression rate of the source is its entropy

rate H

– Theorem 2. Maximum reliable rate over the channel is its mutual information I

– Theorem 3. End-to-end reliable communication happens if and only if H < I, i.e. there is no loss in performance by using a digital interface between source and channel coding

• Impacts of Shannon’s results– after almost 70 years, all communication systems are designed based

on the principles of information theory

– the limits not only serve as benchmarks for evaluating communication schemes, but also provide insights on designing good ones

– basic information theoretic limits in Shannon’s theorems have now been successfully achieved using efficient algorithms and codes

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ENTROPY

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Definition

• Entropy is a measure of the uncertainty of a r.v.

• Consider discrete r.v. X with alphabet and p.m.f.

– log is to the base 2, and entropy is expressed in bits• e.g., the entropy of a fair coin toss is 1 bit

– define , since• adding terms of zero probability does not change the entropy

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( ) Pr[ ], p x X x x

log 0 as 0x x x 0log 0 0

Properties

– entropy is nonnegative

– base of log can be changed

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Example

– H(X)=1 bit when p=0.5• maximum uncertainty

– H(X)=0 bit when p=0 or 1• minimum uncertainty

– concave function of p

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Example

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JOINT ENTROPY AND CONDITIONAL ENTROPY

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Joint Entropy

• Joint entropy is a measure of the uncertainty of a pair of r.v.s

• Consider a pair of discrete r.v.s (X,Y) with alphabet and p.m.f.s

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，

( ) Pr[ ], ( ) Pr[ ], p x X x x p y Y y y ，

Conditional Entropy

• Conditional entropy of a r.v. (Y) given another r.v. (X)

– expected value of entropies of conditional distributions, averaged over conditioning r.v.

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Chain Rule

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Chain Rule

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Example

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Example

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RELATIVE ENTROPY AND MUTUAL INFORMATION

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Relative Entropy

• Relative entropy is a measure of the “distance” between two distributions

– convention:

– if there is any

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0 00log 0, 0 log 0 and log

0 0

pp

q

such that ( ) 0 and ( ) 0, then ( || ) .x p x q x D p q

Example

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Mutual Information

• Mutual information is a measure of the amount of information that one r.v. contains about another r.v.

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RELATIONSHIP BETWEEN ENTROPY AND MUTUAL INFORMATION

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Relation

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Proof

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Illustration

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CHAIN RULES FOR ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

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Chain Rule for Entropy

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Proof

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Alternative Proof

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Chain Rule for Information

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Proof

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Chain Rule for Relative Entropy

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Proof

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JENSEN'S INEQUALITY AND ITS CONSEQUENCES

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Convex & Concave Functions

• Examples:

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2convex functions: , | |, , log (for 0)xx x e x x x

concave functions: log and (for 0)x x x

linear functions are both convex and concaveax b

Convex & Concave Functions

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Jensen’s Inequality

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Information Inequality

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Proof

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Nonnegativity of Mutual Information

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Max. Entropy Dist. – Uniform Dist.

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Conditioning Reduces Entropy

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Independence Bound on Entropy

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Summary

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Summary

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Summary

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[email protected]/Personal/yingcui

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