1896 1920 1987 2006 Computing and Communications 2. Information Theory -Entropy Ying Cui Department of Electronic Engineering Shanghai Jiao Tong University, China 2017, Autumn 1
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 2
Reference • Elements of information theory, T. M. Cover and J. A. Thomas, Wiley 3
OVERVIEW 4
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 5
Information Theory • Information theory has fundamental contributions to other fields 6
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 7
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 8
ENTROPY 9
Definition • Entropy is a measure of the uncertainty of a r.v. • Consider discrete r.v. X with alphabet and p.m.f. ( ) Pr[ ], p x X x x – 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 log 0 as 0 0log0 0 x x x • adding terms of zero probability does not change the entropy 10
Properties – entropy is nonnegative – base of log can be changed 11
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 12
Example 13
JOINT ENTROPY AND CONDITIONAL ENTROPY 14
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 ( ) Pr[ ], ( ) Pr[ ], p x X x x p y Y y y 15
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. 16
Chain Rule 17
Chain Rule 18
Example 19
Example 20
RELATIVE ENTROPY AND MUTUAL INFORMATION 21
Relative Entropy • Relative entropy is a measure of the “distance” between two distributions 0 0 p – convention: 0log 0, 0log 0 and log p 0 0 q – if there is any such that ( ) 0 and ( ) 0, then ( || ) . x p x q x D p q 22
Example 23
Mutual Information • Mutual information is a measure of the amount of information that one r.v. contains about another r.v. 24
RELATIONSHIP BETWEEN ENTROPY AND MUTUAL INFORMATION 25
Relation 26
Proof 27
Illustration 28
CHAIN RULES FOR ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION 29
Chain Rule for Entropy 30
Proof 31
Alternative Proof 32
Chain Rule for Information 33
Proof 34
Chain Rule for Relative Entropy 35
Proof 36
JENSEN'S INEQUALITY AND ITS CONSEQUENCES 37
Convex & Concave Functions • Examples: x 2 x convex functions: , | |, , log (for 0) x x e x x x concave functions: log and (for 0) x x linear functions are both convex and concave ax b 38
Convex & Concave Functions 39
Jensen’s Inequality 40
Information Inequality 41
Proof 42
Nonnegativity of Mutual Information 43
Max. Entropy Dist. – Uniform Dist. 44
Conditioning Reduces Entropy 45
Independence Bound on Entropy 46
Summary 47
Summary 48
Summary 49
cuiying@sjtu.edu.cn iwct.sjtu.edu.cn/Personal/yingcui 50
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