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ELEMENTS OF INFORMATION THEORY
Second Edition
THOMAS M. COVER JOY A. THOMAS
ffiWILEY-
~INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION

ELEMENTS OF INFORMATION THEORY

ELEMENTS OF INFORMATION THEORY
Second Edition
THOMAS M. COVER JOY A. THOMAS
ffiWILEY-
~INTERSCIENCE
A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2006 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Cover, T. M., 1938– Elements of information theory/by Thomas M. Cover, Joy A. Thomas.–2nd ed. p. cm. “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN-13 978-0-471-24195-9 ISBN-10 0-471-24195-4 1. Information theory. I. Thomas, Joy A. II. Title.
Q360.C68 2005 003 .54–dc22
2005047799
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1

CONTENTS

Contents

v

Preface to the Second Edition

xv

Preface to the First Edition

xvii

Acknowledgments for the Second Edition

xxi

Acknowledgments for the First Edition

xxiii

1 Introduction and Preview

1

1.1 Preview of the Book 5

2 Entropy, Relative Entropy, and Mutual Information

13

2.1 Entropy 13

2.2 Joint Entropy and Conditional Entropy 16

2.3 Relative Entropy and Mutual Information 19

2.4 Relationship Between Entropy and Mutual Information 20

2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information 22

2.6 Jensen’s Inequality and Its Consequences 25

2.7 Log Sum Inequality and Its Applications 30

2.8 Data-Processing Inequality 34

2.9 Sufficient Statistics 35

2.10 Fano’s Inequality 37

Summary 41

Problems 43

Historical Notes 54

v

vi CONTENTS

3 Asymptotic Equipartition Property

57

3.1 Asymptotic Equipartition Property Theorem 58

3.2 Consequences of the AEP: Data Compression 60

3.3 High-Probability Sets and the Typical Set 62

Summary 64

Problems 64

Historical Notes 69

4 Entropy Rates of a Stochastic Process

71

4.1 Markov Chains 71

4.2 Entropy Rate 74

4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph 78

4.4 Second Law of Thermodynamics 81

4.5 Functions of Markov Chains 84

Summary 87

Problems 88

Historical Notes 100

5 Data Compression

103

5.1 Examples of Codes 103

5.2 Kraft Inequality 107

5.3 Optimal Codes 110

5.4 Bounds on the Optimal Code Length 112

5.5 Kraft Inequality for Uniquely Decodable Codes 115

5.6 Huffman Codes 118

5.7 Some Comments on Huffman Codes 120

5.8 Optimality of Huffman Codes 123

5.9 Shannon–Fano–Elias Coding 127

5.10 Competitive Optimality of the Shannon Code 130

5.11 Generation of Discrete Distributions from Fair Coins 134

Summary 141

Problems 142

Historical Notes 157

CONTENTS vii

6 Gambling and Data Compression

159

6.1 The Horse Race 159

6.2 Gambling and Side Information 164

6.3 Dependent Horse Races and Entropy Rate 166

6.4 The Entropy of English 168

6.5 Data Compression and Gambling 171

6.6 Gambling Estimate of the Entropy of English 173

Summary 175

Problems 176

Historical Notes 182

7 Channel Capacity

183

7.1 Examples of Channel Capacity 184

7.1.1 Noiseless Binary Channel 184

7.1.2 Noisy Channel with Nonoverlapping Outputs 185

7.1.3 Noisy Typewriter 186

7.1.4 Binary Symmetric Channel 187

7.1.5 Binary Erasure Channel 188

7.2 Symmetric Channels 189

7.3 Properties of Channel Capacity 191

7.4 Preview of the Channel Coding Theorem 191

7.5 Definitions 192

7.6 Jointly Typical Sequences 195

7.7 Channel Coding Theorem 199

7.8 Zero-Error Codes 205

7.9 Fano’s Inequality and the Converse to the Coding Theorem 206

7.10 Equality in the Converse to the Channel Coding Theorem 208

7.11 Hamming Codes 210

7.12 Feedback Capacity 216

7.13 Source–Channel Separation Theorem 218

Summary 222

Problems 223

Historical Notes 240

viii CONTENTS

8 Differential Entropy

243

8.1 Definitions 243

8.2 AEP for Continuous Random Variables 245

8.3 Relation of Differential Entropy to Discrete Entropy 247

8.4 Joint and Conditional Differential Entropy 249

8.5 Relative Entropy and Mutual Information 250

8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information 252

Summary 256

Problems 256

Historical Notes 259

9 Gaussian Channel

261

9.1 Gaussian Channel: Definitions 263

9.2 Converse to the Coding Theorem for Gaussian

Channels 268

9.3 Bandlimited Channels 270

9.4 Parallel Gaussian Channels 274

9.5 Channels with Colored Gaussian Noise 277

9.6 Gaussian Channels with Feedback 280

Summary 289

Problems 290

Historical Notes 299

10 Rate Distortion Theory

301

10.1 Quantization 301

10.2 Definitions 303

10.3 Calculation of the Rate Distortion Function 307

10.3.1 Binary Source 307

10.3.2 Gaussian Source 310

10.3.3 Simultaneous Description of Independent Gaussian Random Variables 312

10.4 Converse to the Rate Distortion Theorem 315

10.5 Achievability of the Rate Distortion Function 318

10.6 Strongly Typical Sequences and Rate Distortion 325

10.7 Characterization of the Rate Distortion Function 329

CONTENTS ix

10.8 Computation of Channel Capacity and the Rate Distortion Function 332
Summary 335 Problems 336 Historical Notes 345

11 Information Theory and Statistics

347

11.1 Method of Types 347

11.2 Law of Large Numbers 355

11.3 Universal Source Coding 357

11.4 Large Deviation Theory 360

11.5 Examples of Sanov’s Theorem 364

11.6 Conditional Limit Theorem 366

11.7 Hypothesis Testing 375

11.8 Chernoff–Stein Lemma 380

11.9 Chernoff Information 384

11.10 Fisher Information and the Crame´r–Rao Inequality 392

Summary 397

Problems 399

Historical Notes 408

12 Maximum Entropy

409

12.1 Maximum Entropy Distributions 409

12.2 Examples 411

12.3 Anomalous Maximum Entropy Problem 413

12.4 Spectrum Estimation 415

12.5 Entropy Rates of a Gaussian Process 416

12.6 Burg’s Maximum Entropy Theorem 417

Summary 420

Problems 421

Historical Notes 425

13 Universal Source Coding

427

13.1 Universal Codes and Channel Capacity 428

13.2 Universal Coding for Binary Sequences 433

13.3 Arithmetic Coding 436

x CONTENTS

13.4 Lempel–Ziv Coding 440 13.4.1 Sliding Window Lempel–Ziv Algorithm 441 13.4.2 Tree-Structured Lempel–Ziv Algorithms 442
13.5 Optimality of Lempel–Ziv Algorithms 443 13.5.1 Sliding Window Lempel–Ziv Algorithms 443 13.5.2 Optimality of Tree-Structured Lempel–Ziv Compression 448
Summary 456 Problems 457 Historical Notes 461

14 Kolmogorov Complexity

463

14.1 Models of Computation 464

14.2 Kolmogorov Complexity: Definitions and Examples 466

14.3 Kolmogorov Complexity and Entropy 473

14.4 Kolmogorov Complexity of Integers 475

14.5 Algorithmically Random and Incompressible Sequences 476

14.6 Universal Probability 480

14.7 Kolmogorov complexity 482

14.8

484

14.9 Universal Gambling 487

14.10 Occam’s Razor 488

14.11 Kolmogorov Complexity and Universal Probability 490

14.12 Kolmogorov Sufficient Statistic 496

14.13 Minimum Description Length Principle 500

Summary 501

Problems 503

Historical Notes 507

15 Network Information Theory

509

15.1 Gaussian Multiple-User Channels 513

CONTENTS xi
15.1.1 Single-User Gaussian Channel 513 15.1.2 Gaussian Multiple-Access Channel
with m Users 514 15.1.3 Gaussian Broadcast Channel 515 15.1.4 Gaussian Relay Channel 516 15.1.5 Gaussian Interference Channel 518 15.1.6 Gaussian Two-Way Channel 519 15.2 Jointly Typical Sequences 520 15.3 Multiple-Access Channel 524 15.3.1 Achievability of the Capacity Region for the
Multiple-Access Channel 530 15.3.2 Comments on the Capacity Region for the
Multiple-Access Channel 532 15.3.3 Convexity of the Capacity Region of the
Multiple-Access Channel 534 15.3.4 Converse for the Multiple-Access
Channel 538 15.3.5 m-User Multiple-Access Channels 543 15.3.6 Gaussian Multiple-Access Channels 544 15.4 Encoding of Correlated Sources 549 15.4.1 Achievability of the Slepian–Wolf
Theorem 551 15.4.2 Converse for the Slepian–Wolf
Theorem 555 15.4.3 Slepian–Wolf Theorem for Many
Sources 556 15.4.4 Interpretation of Slepian–Wolf
Coding 557 15.5 Duality Between Slepian–Wolf Encoding and
Multiple-Access Channels 558 15.6 Broadcast Channel 560
15.6.1 Definitions for a Broadcast Channel 563 15.6.2 Degraded Broadcast Channels 564 15.6.3 Capacity Region for the Degraded Broadcast
Channel 565 15.7 Relay Channel 571 15.8 Source Coding with Side Information 575 15.9 Rate Distortion with Side Information 580

xii CONTENTS

15.10 General Multiterminal Networks 587 Summary 594 Problems 596 Historical Notes 609

16 Information Theory and Portfolio Theory

613

16.1 The Stock Market: Some Definitions 613

16.2 Kuhn–Tucker Characterization of the Log-Optimal Portfolio 617

16.3 Asymptotic Optimality of the Log-Optimal Portfolio 619

16.4 Side Information and the Growth Rate 621

16.5 Investment in Stationary Markets 623

16.6 Competitive Optimality of the Log-Optimal Portfolio 627

16.7 Universal Portfolios 629

16.7.1 Finite-Horizon Universal Portfolios 631

16.7.2 Horizon-Free Universal Portfolios 638

16.8 Shannon–McMillan–Breiman Theorem (General AEP) 644

Summary 650

Problems 652

Historical Notes 655

17 Inequalities in Information Theory

657

17.1 Basic Inequalities of Information Theory 657

17.2 Differential Entropy 660

17.3 Bounds on Entropy and Relative Entropy 663

17.4 Inequalities for Types 665

17.5 Combinatorial Bounds on Entropy 666

17.6 Entropy Rates of Subsets 667

17.7 Entropy and Fisher Information 671

17.8 Entropy Power Inequality and Brunn–Minkowski Inequality 674

17.9 Inequalities for Determinants 679

CONTENTS xiii
17.10 Inequalities for Ratios of Determinants 683 Summary 686 Problems 686 Historical Notes 687

Bibliography

689

List of Symbols

723

Index

727

PREFACE TO THE SECOND EDITION
In the years since the publication of the first edition, there were many aspects of the book that we wished to improve, to rearrange, or to expand, but the constraints of reprinting would not allow us to make those changes between printings. In the new edition, we now get a chance to make some of these changes, to add problems, and to discuss some topics that we had omitted from the first edition.
The key changes include a reorganization of the chapters to make the book easier to teach, and the addition of more than two hundred new problems. We have added material on universal portfolios, universal source coding, Gaussian feedback capacity, network information theory, and developed the duality of data compression and channel capacity. A new chapter has been added and many proofs have been simplified. We have also updated the references and historical notes.
The material in this book can be taught in a two-quarter sequence. The first quarter might cover Chapters 1 to 9, which includes the asymptotic equipartition property, data compression, and channel capacity, culminating in the capacity of the Gaussian channel. The second quarter could cover the remaining chapters, including rate distortion, the method of types, Kolmogorov complexity, network information theory, universal source coding, and portfolio theory. If only one semester is available, we would add rate distortion and a single lecture each on Kolmogorov complexity and network information theory to the first semester. A web site, http://www.elementsofinformationtheory.com, provides links to additional material and solutions to selected problems.
In the years since the first edition of the book, information theory celebrated its 50th birthday (the 50th anniversary of Shannon’s original paper that started the field), and ideas from information theory have been applied to many problems of science and technology, including bioinformatics, web search, wireless communication, video compression, and
xv

xvi PREFACE TO THE SECOND EDITION
others. The list of applications is endless, but it is the elegance of the fundamental mathematics that is still the key attraction of this area. We hope that this book will give some insight into why we believe that this is one of the most interesting areas at the intersection of mathematics, physics, statistics, and engineering.
Tom Cover Joy Thomas
Palo Alto, California January 2006

PREFACE TO THE FIRST EDITION
This is intended to be a simple and accessible book on information theory. As Einstein said, “Everything should be made as simple as possible, but no simpler.” Although we have not verified the quote (first found in a fortune cookie), this point of view drives our development throughout the book. There are a few key ideas and techniques that, when mastered, make the subject appear simple and provide great intuition on new questions.
This book has arisen from over ten years of lectures in a two-quarter sequence of a senior and first-year graduate-level course in information theory, and is intended as an introduction to information theory for students of communication theory, computer science, and statistics.
There are two points to be made about the simplicities inherent in information theory. First, certain quantities like entropy and mutual information arise as the answers to fundamental questions. For example, entropy is the minimum descriptive complexity of a random variable, and mutual information is the communication rate in the presence of noise. Also, as we shall point out, mutual information corresponds to the increase in the doubling rate of wealth given side information. Second, the answers to information theoretic questions have a natural algebraic structure. For example, there is a chain rule for entropies, and entropy and mutual information are related. Thus the answers to problems in data compression and communication admit extensive interpretation. We all know the feeling that follows when one investigates a problem, goes through a large amount of algebra, and finally investigates the answer to find that the entire problem is illuminated not by the analysis but by the inspection of the answer. Perhaps the outstanding examples of this in physics are Newton’s laws and Schro¨dinger’s wave equation. Who could have foreseen the awesome philosophical interpretations of Schro¨dinger’s wave equation?
In the text we often investigate properties of the answer before we look at the question. For example, in Chapter 2, we define entropy, relative entropy, and mutual information and study the relationships and a few
xvii

xviii PREFACE TO THE FIRST EDITION
interpretations of them, showing how the answers fit together in various ways. Along the way we speculate on the meaning of the second law of thermodynamics. Does entropy always increase? The answer is yes and no. This is the sort of result that should please experts in the area but might be overlooked as standard by the novice.
In fact, that brings up a point that often occurs in teaching. It is fun to find new proofs or slightly new results that no one else knows. When one presents these ideas along with the established material in class, the response is “sure, sure, sure.” But the excitement of teaching the material is greatly enhanced. Thus we have derived great pleasure from investigating a number of new ideas in this textbook.
Examples of some of the new material in this text include the chapter on the relationship of information theory to gambling, the work on the universality of the second law of thermodynamics in the context of Markov chains, the joint typicality proofs of the channel capacity theorem, the competitive optimality of Huffman codes, and the proof of Burg’s theorem on maximum entropy spectral density estimation. Also, the chapter on Kolmogorov complexity has no counterpart in other information theory texts. We have also taken delight in relating Fisher information, mutual information, the central limit theorem, and the Brunn–Minkowski and entropy power inequalities. To our surprise, many of the classical results on determinant inequalities are most easily proved using information theoretic inequalities.
Even though the field of information theory has grown considerably since Shannon’s original paper, we have strived to emphasize its coherence. While it is clear that Shannon was motivated by problems in communication theory when he developed information theory, we treat information theory as a field of its own with applications to communication theory and statistics. We were drawn to the field of information theory from backgrounds in communication theory, probability theory, and statistics, because of the apparent impossibility of capturing the intangible concept of information.
Since most of the results in the book are given as theorems and proofs, we expect the elegance of the results to speak for themselves. In many cases we actually describe the properties of the solutions before the problems. Again, the properties are interesting in themselves and provide a natural rhythm for the proofs that follow.
One innovation in the presentation is our use of long chains of inequalities with no intervening text followed immediately by the explanations. By the time the reader comes to many of these proofs, we expect that he or she will be able to follow most of these steps without any explanation and will be able to pick out the needed explanations. These chains of

PREFACE TO THE FIRST EDITION xix
inequalities serve as pop quizzes in which the reader can be reassured of having the knowledge needed to prove some important theorems. The natural flow of these proofs is so compelling that it prompted us to flout one of the cardinal rules of technical writing; and the absence of verbiage makes the logical necessity of the ideas evident and the key ideas perspicuous. We hope that by the end of the book the reader will share our appreciation of the elegance, simplicity, and naturalness of information theory.
Throughout the book we use the method of weakly typical sequences, which has its origins in Shannon’s original 1948 work but was formally developed in the early 1970s. The key idea here is the asymptotic equipartition property, which can be roughly paraphrased as “Almost everything is almost equally probable.”
Chapter 2 includes the basic algebraic relationships of entropy, relative entropy, and mutual information. The asymptotic equipartition property (AEP) is given central prominence in Chapter 3. This leads us to discuss the entropy rates of stochastic processes and data compression in Chapters 4 and 5. A gambling sojourn is taken in Chapter 6, where the duality of data compression and the growth rate of wealth is developed.
The sensational success of Kolmogorov complexity as an intellectual foundation for information theory is explored in Chapter 14. Here we replace the goal of finding a description that is good on the average with the goal of finding the universally shortest description. There is indeed a universal notion of the descriptive complexity of an object. Here also the wonderful number is investigated. This number, which is the binary expansion of the probability that a Turing machine will halt, reveals many of the secrets of mathematics.
Channel capacity is established in Chapter 7. The necessary material on differential entropy is developed in Chapter 8, laying the groundwork for the extension of previous capacity theorems to continuous noise channels. The capacity of the fundamental Gaussian channel is investigated in Chapter 9.
The relationship between information theory and statistics, first studied by Kullback in the early 1950s and relatively neglected since, is developed in Chapter 11. Rate distortion theory requires a little more background than its noiseless data compression counterpart, which accounts for its placement as late as Chapter 10 in the text.
The huge subject of network information theory, which is the study of the simultaneously achievable flows of information in the presence of noise and interference, is developed in Chapter 15. Many new ideas come into play in network information theory. The primary new ingredients are interference and feedback. Chapter 16 considers the stock market, which is

xx PREFACE TO THE FIRST EDITION
the generalization of the gambling processes considered in Chapter 6, and shows again the close correspondence of information theory and gambling.
Chapter 17, on inequalities in information theory, gives us a chance to recapitulate the interesting inequalities strewn throughout the book, put them in a new framework, and then add some interesting new inequalities on the entropy rates of randomly drawn subsets. The beautiful relationship of the Brunn–Minkowski inequality for volumes of set sums, the entropy power inequality for the effective variance of the sum of independent random variables, and the Fisher information inequalities are made explicit here.
We have made an attempt to keep the theory at a consistent level. The mathematical level is a reasonably high one, probably the senior or first-year graduate level, with a background of at least one good semester course in probability and a solid background in mathematics. We have, however, been able to avoid the use of measure theory. Measure theory comes up only briefly in the proof of the AEP for ergodic processes in Chapter 16. This fits in with our belief that the fundamentals of information theory are orthogonal to the techniques required to bring them to their full generalization.
The essential vitamins are contained in Chapters 2, 3, 4, 5, 7, 8, 9, 11, 10, and 15. This subset of chapters can be read without essential reference to the others and makes a good core of understanding. In our opinion, Chapter 14 on Kolmogorov complexity is also essential for a deep understanding of information theory. The rest, ranging from gambling to inequalities, is part of the terrain illuminated by this coherent and beautiful subject.
Every course has its first lecture, in which a sneak preview and overview of ideas is presented. Chapter 1 plays this role.
Tom Cover Joy Thomas
Palo Alto, California June 1990

ACKNOWLEDGMENTS FOR THE SECOND EDITION
Since the appearance of the first edition, we have been fortunate to receive feedback, suggestions, and corrections from a large number of readers. It would be impossible to thank everyone who has helped us in our efforts, but we would like to list some of them. In particular, we would like to thank all the faculty who taught courses based on this book and the students who took those courses; it is through them that we learned to look at the same material from a different perspective.
In particular, we would like to thank Andrew Barron, Alon Orlitsky, T. S. Han, Raymond Yeung, Nam Phamdo, Franz Willems, and Marty Cohn for their comments and suggestions. Over the years, students at Stanford have provided ideas and inspirations for the changes—these include George Gemelos, Navid Hassanpour, Young-Han Kim, Charles Mathis, Styrmir Sigurjonsson, Jon Yard, Michael Baer, Mung Chiang, Suhas Diggavi, Elza Erkip, Paul Fahn, Garud Iyengar, David Julian, Yiannis Kontoyiannis, Amos Lapidoth, Erik Ordentlich, Sandeep Pombra, Jim Roche, Arak Sutivong, Joshua Sweetkind-Singer, and Assaf Zeevi. Denise Murphy provided much support and help during the preparation of the second edition.
Joy Thomas would like to acknowledge the support of colleagues at IBM and Stratify who provided valuable comments and suggestions. Particular thanks are due Peter Franaszek, C. S. Chang, Randy Nelson, Ramesh Gopinath, Pandurang Nayak, John Lamping, Vineet Gupta, and Ramana Venkata. In particular, many hours of dicussion with Brandon Roy helped refine some of the arguments in the book. Above all, Joy would like to acknowledge that the second edition would not have been possible without the support and encouragement of his wife, Priya, who makes all things worthwhile.
Tom Cover would like to thank his students and his wife, Karen.
xxi

ACKNOWLEDGMENTS FOR THE FIRST EDITION
We wish to thank everyone who helped make this book what it is. In particular, Aaron Wyner, Toby Berger, Masoud Salehi, Alon Orlitsky, Jim Mazo and Andrew Barron have made detailed comments on various drafts of the book which guided us in our final choice of content. We would like to thank Bob Gallager for an initial reading of the manuscript and his encouragement to publish it. Aaron Wyner donated his new proof with Ziv on the convergence of the Lempel-Ziv algorithm. We would also like to thank Normam Abramson, Ed van der Meulen, Jack Salz and Raymond Yeung for their suggested revisions.
Certain key visitors and research associates contributed as well, including Amir Dembo, Paul Algoet, Hirosuke Yamamoto, Ben Kawabata, M. Shimizu and Yoichiro Watanabe. We benefited from the advice of John Gill when he used this text in his class. Abbas El Gamal made invaluable contributions, and helped begin this book years ago when we planned to write a research monograph on multiple user information theory. We would also like to thank the Ph.D. students in information theory as this book was being written: Laura Ekroot, Will Equitz, Don Kimber, Mitchell Trott, Andrew Nobel, Jim Roche, Erik Ordentlich, Elza Erkip and Vittorio Castelli. Also Mitchell Oslick, Chien-Wen Tseng and Michael Morrell were among the most active students in contributing questions and suggestions to the text. Marc Goldberg and Anil Kaul helped us produce some of the figures. Finally we would like to thank Kirsten Goodell and Kathy Adams for their support and help in some of the aspects of the preparation of the manuscript.
Joy Thomas would also like to thank Peter Franaszek, Steve Lavenberg, Fred Jelinek, David Nahamoo and Lalit Bahl for their encouragment and support during the final stages of production of this book.
xxiii

CHAPTER 1
INTRODUCTION AND PREVIEW
Information theory answers two fundamental questions in communication theory: What is the ultimate data compression (answer: the entropy H ), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam’s Razor: “The simplest explanation is best”), and to probability and statistics (error exponents for optimal hypothesis testing and estimation).
This “first lecture” chapter goes backward and forward through information theory and its naturally related ideas. The full definitions and study of the subject begin in Chapter 2. Figure 1.1 illustrates the relationship of information theory to other fields. As the figure suggests, information theory intersects physics (statistical mechanics), mathematics (probability theory), electrical engineering (communication theory), and computer science (algorithmic complexity). We now describe the areas of intersection in greater detail.
Electrical Engineering (Communication Theory). In the early 1940s it was thought to be impossible to send information at a positive rate with negligible probability of error. Shannon surprised the communication theory community by proving that the probability of error could be made nearly zero for all communication rates below channel capacity. The capacity can be computed simply from the noise characteristics of the channel. Shannon further argued that random processes such as music and speech have an irreducible complexity below which the signal cannot be compressed. This he named the entropy, in deference to the parallel use of this word in thermodynamics, and argued that if the entropy of the
Elements of Information Theory, Second Edition, By Thomas M. Cover and Joy A. Thomas Copyright  2006 John Wiley & Sons, Inc.
1

2 INTRODUCTION AND PREVIEW

CommTuhneiocraytion CoLmimmTituhsneoiocfraytion

PTrhoeboarbyility DeTvLihaaetrioLgorienmesmit s

Information Theory

Physics T IhnefrTQohrumeaomAondratEtyyiunPoamnmics Inequalities Mathematics

Statistics HyInTpfFeooistrshmthieneasgrtiison
KoClmoomgpolreoxvity ComScpiuetnecre

Portfolio Theory Kelly Gambling
Economics

FIGURE 1.1. Relationship of information theory to other fields.

Data compression limit

Data transmission limit

min l (X; X^ )

max l(X; Y )

FIGURE 1.2. Information theory as the extreme points of communication theory.

source is less than the capacity of the channel, asymptotically error-free communication can be achieved.
Information theory today represents the extreme points of the set of all possible communication schemes, as shown in the fanciful Figure 1.2. The data compression minimum I (X; Xˆ ) lies at one extreme of the set of communication ideas. All data compression schemes require description

INTRODUCTION AND PREVIEW 3
rates at least equal to this minimum. At the other extreme is the data transmission maximum I (X; Y ), known as the channel capacity. Thus, all modulation schemes and data compression schemes lie between these limits.
Information theory also suggests means of achieving these ultimate limits of communication. However, these theoretically optimal communication schemes, beautiful as they are, may turn out to be computationally impractical. It is only because of the computational feasibility of simple modulation and demodulation schemes that we use them rather than the random coding and nearest-neighbor decoding rule suggested by Shannon’s proof of the channel capacity theorem. Progress in integrated circuits and code design has enabled us to reap some of the gains suggested by Shannon’s theory. Computational practicality was finally achieved by the advent of turbo codes. A good example of an application of the ideas of information theory is the use of error-correcting codes on compact discs and DVDs.
Recent work on the communication aspects of information theory has concentrated on network information theory: the theory of the simultaneous rates of communication from many senders to many receivers in the presence of interference and noise. Some of the trade-offs of rates between senders and receivers are unexpected, and all have a certain mathematical simplicity. A unifying theory, however, remains to be found.
Computer Science (Kolmogorov Complexity). Kolmogorov, Chaitin, and Solomonoff put forth the idea that the complexity of a string of data can be defined by the length of the shortest binary computer program for computing the string. Thus, the complexity is the minimal description length. This definition of complexity turns out to be universal, that is, computer independent, and is of fundamental importance. Thus, Kolmogorov complexity lays the foundation for the theory of descriptive complexity. Gratifyingly, the Kolmogorov complexity K is approximately equal to the Shannon entropy H if the sequence is drawn at random from a distribution that has entropy H . So the tie-in between information theory and Kolmogorov complexity is perfect. Indeed, we consider Kolmogorov complexity to be more fundamental than Shannon entropy. It is the ultimate data compression and leads to a logically consistent procedure for inference.
There is a pleasing complementary relationship between algorithmic complexity and computational complexity. One can think about computational complexity (time complexity) and Kolmogorov complexity (program length or descriptive complexity) as two axes corresponding to

4 INTRODUCTION AND PREVIEW
program running time and program length. Kolmogorov complexity focuses on minimizing along the second axis, and computational complexity focuses on minimizing along the first axis. Little work has been done on the simultaneous minimization of the two.
Physics (Thermodynamics). Statistical mechanics is the birthplace of entropy and the second law of thermodynamics. Entropy always increases. Among other things, the second law allows one to dismiss any claims to perpetual motion machines. We discuss the second law briefly in Chapter 4.
Mathematics (Probability Theory and Statistics). The fundamental quantities of information theory—entropy, relative entropy, and mutual information—are defined as functionals of probability distributions. In turn, they characterize the behavior of long sequences of random variables and allow us to estimate the probabilities of rare events (large deviation theory) and to find the best error exponent in hypothesis tests.
Philosophy of Science (Occam’s Razor). William of Occam said “Causes shall not be multiplied beyond necessity,” or to paraphrase it, “The simplest explanation is best.” Solomonoff and Chaitin argued persuasively that one gets a universally good prediction procedure if one takes a weighted combination of all programs that explain the data and observes what they print next. Moreover, this inference will work in many problems not handled by statistics. For example, this procedure will eventually predict the subsequent digits of π . When this procedure is applied to coin flips that come up heads with probability 0.7, this too will be inferred. When applied to the stock market, the procedure should essentially find all the “laws” of the stock market and extrapolate them optimally. In principle, such a procedure would have found Newton’s laws of physics. Of course, such inference is highly impractical, because weeding out all computer programs that fail to generate existing data will take impossibly long. We would predict what happens tomorrow a hundred years from now.
Economics (Investment). Repeated investment in a stationary stock market results in an exponential growth of wealth. The growth rate of the wealth is a dual of the entropy rate of the stock market. The parallels between the theory of optimal investment in the stock market and information theory are striking. We develop the theory of investment to explore this duality.
Computation vs. Communication. As we build larger computers out of smaller components, we encounter both a computation limit and a communication limit. Computation is communication limited and communication is computation limited. These become intertwined, and thus

1.1 PREVIEW OF THE BOOK 5
all of the developments in communication theory via information theory should have a direct impact on the theory of computation.

1.1 PREVIEW OF THE BOOK

The initial questions treated by information theory lay in the areas of data compression and transmission. The answers are quantities such as entropy and mutual information, which are functions of the probability distributions that underlie the process of communication. A few definitions will aid the initial discussion. We repeat these definitions in Chapter 2.
The entropy of a random variable X with a probability mass function p(x) is defined by

H (X) = − p(x) log2 p(x).

(1.1)

x

We use logarithms to base 2. The entropy will then be measured in bits. The entropy is a measure of the average uncertainty in the random variable. It is the number of bits on average required to describe the random variable.

Example 1.1.1 Consider a random variable that has a uniform distribution over 32 outcomes. To identify an outcome, we need a label that takes on 32 different values. Thus, 5-bit strings suffice as labels.
The entropy of this random variable is

32

32

H (X) = − p(i) log p(i) = −

1 log 1 = log 32 = 5 bits,

32 32

i=1

i=1

(1.2)

which agrees with the number of bits needed to describe X. In this case,

all the outcomes have representations of the same length.

Now consider an example with nonuniform distribution.

Example 1.1.2 Suppose that we have a horse race with eight horses

taking part. Assume that the probabilities of winning for the eight horses

are

1 2

,

1 4

,

1 8

,

1 16

,

1 64

,

1 64

,

1 64

,

1 64

.

We

can

calculate

the

entropy

of

the

horse

race as

H (X) = − 1 log 1 − 1 log 1 − 1 log 1 − 1 log 1 − 4 1 log 1 2 2 4 4 8 8 16 16 64 64

= 2 bits.

(1.3)

6 INTRODUCTION AND PREVIEW
Suppose that we wish to send a message indicating which horse won the race. One alternative is to send the index of the winning horse. This description requires 3 bits for any of the horses. But the win probabilities are not uniform. It therefore makes sense to use shorter descriptions for the more probable horses and longer descriptions for the less probable ones, so that we achieve a lower average description length. For example, we could use the following set of bit strings to represent the eight horses: 0, 10, 110, 1110, 111100, 111101, 111110, 111111. The average description length in this case is 2 bits, as opposed to 3 bits for the uniform code. Notice that the average description length in this case is equal to the entropy. In Chapter 5 we show that the entropy of a random variable is a lower bound on the average number of bits required to represent the random variable and also on the average number of questions needed to identify the variable in a game of “20 questions.” We also show how to construct representations that have an average length within 1 bit of the entropy.
The concept of entropy in information theory is related to the concept of entropy in statistical mechanics. If we draw a sequence of n independent and identically distributed (i.i.d.) random variables, we will show that the probability of a “typical” sequence is about 2−nH(X) and that there are about 2nH(X) such typical sequences. This property [known as the asymptotic equipartition property (AEP)] is the basis of many of the proofs in information theory. We later present other problems for which entropy arises as a natural answer (e.g., the number of fair coin flips needed to generate a random variable).
The notion of descriptive complexity of a random variable can be extended to define the descriptive complexity of a single string. The Kolmogorov complexity of a binary string is defined as the length of the shortest computer program that prints out the string. It will turn out that if the string is indeed random, the Kolmogorov complexity is close to the entropy. Kolmogorov complexity is a natural framework in which to consider problems of statistical inference and modeling and leads to a clearer understanding of Occam’s Razor: “The simplest explanation is best.” We describe some simple properties of Kolmogorov complexity in Chapter 1.
Entropy is the uncertainty of a single random variable. We can define conditional entropy H (X|Y ), which is the entropy of a random variable conditional on the knowledge of another random variable. The reduction in uncertainty due to another random variable is called the mutual information. For two random variables X and Y this reduction is the mutual

1.1 PREVIEW OF THE BOOK 7

information

I (X; Y ) = H (X) − H (X|Y ) = p(x, y) log p(x, y) . (1.4)

x,y

p(x)p(y)

The mutual information I (X; Y ) is a measure of the dependence between the two random variables. It is symmetric in X and Y and always nonnegative and is equal to zero if and only if X and Y are independent.
A communication channel is a system in which the output depends
probabilistically on its input. It is characterized by a probability transition matrix p(y|x) that determines the conditional distribution of the output given the input. For a communication channel with input X and output Y , we can define the capacity C by

C = max I (X; Y ).
p(x)

(1.5)

Later we show that the capacity is the maximum rate at which we can send information over the channel and recover the information at the output with a vanishingly low probability of error. We illustrate this with a few examples.

Example 1.1.3 (Noiseless binary channel ) For this channel, the binary input is reproduced exactly at the output. This channel is illustrated in Figure 1.3. Here, any transmitted bit is received without error. Hence, in each transmission, we can send 1 bit reliably to the receiver, and the capacity is 1 bit. We can also calculate the information capacity C = max I (X; Y ) = 1 bit.

Example 1.1.4 (Noisy four-symbol channel ) Consider the channel

shown in Figure 1.4. In this channel, each input letter is received either as

the

same

letter

with

probability

1 2

or

as

the

next

letter

with

probability

1 2

.

If we use all four input symbols, inspection of the output would not reveal

with certainty which input symbol was sent. If, on the other hand, we use

0

0

1

1

FIGURE 1.3. Noiseless binary channel. C = 1 bit.

8 INTRODUCTION AND PREVIEW

1

1

2

2

3

3

4

4

FIGURE 1.4. Noisy channel.

only two of the inputs (1 and 3, say), we can tell immediately from the output which input symbol was sent. This channel then acts like the noiseless channel of Example 1.1.3, and we can send 1 bit per transmission over this channel with no errors. We can calculate the channel capacity C = max I (X; Y ) in this case, and it is equal to 1 bit per transmission, in agreement with the analysis above.
In general, communication channels do not have the simple structure of this example, so we cannot always identify a subset of the inputs to send information without error. But if we consider a sequence of transmissions, all channels look like this example and we can then identify a subset of the input sequences (the codewords) that can be used to transmit information over the channel in such a way that the sets of possible output sequences associated with each of the codewords are approximately disjoint. We can then look at the output sequence and identify the input sequence with a vanishingly low probability of error.

Example 1.1.5 (Binary symmetric channel ) This is the basic example of a noisy communication system. The channel is illustrated in Figure 1.5.

1−p

0

0

p p

1

1

1−p

FIGURE 1.5. Binary symmetric channel.

1.1 PREVIEW OF THE BOOK 9

The channel has a binary input, and its output is equal to the input with probability 1 − p. With probability p, on the other hand, a 0 is received as a 1, and vice versa. In this case, the capacity of the channel can be calculated to be C = 1 + p log p + (1 − p) log(1 − p) bits per transmission. However, it is no longer obvious how one can achieve this capacity. If we use the channel many times, however, the channel begins to look like the noisy four-symbol channel of Example 1.1.4, and we can send information at a rate C bits per transmission with an arbitrarily low probability of error.

The ultimate limit on the rate of communication of information over a channel is given by the channel capacity. The channel coding theorem shows that this limit can be achieved by using codes with a long block length. In practical communication systems, there are limitations on the complexity of the codes that we can use, and therefore we may not be able to achieve capacity.
Mutual information turns out to be a special case of a more general quantity called relative entropy D(p||q), which is a measure of the “distance” between two probability mass functions p and q. It is defined as

p(x)

D(p||q) = p(x) log - - -.

x

q(x)

(1.6)

Although relative entropy is not a true metric, it has some of the properties of a metric. In particular, it is always nonnegative and is zero if and only if p = q. Relative entropy arises as the exponent in the probability of error in a hypothesis test between distributions p and q. Relative entropy can be used to define a geometry for probability distributions that allows us to interpret many of the results of large deviation theory.
There are a number of parallels between information theory and the theory of investment in a stock market. A stock market is defined by a random vector X whose elements are nonnegative numbers equal to the ratio of the price of a stock at the end of a day to the price at the beginning of the day. For a stock market with distribution F (x), we can define the doubling rate W as

W = max

log bt x dF (x).

b:bi ≥0, bi =1

(1.7)

The doubling rate is the maximum asymptotic exponent in the growth of wealth. The doubling rate has a number of properties that parallel the properties of entropy. We explore some of these properties in Chapter 16.

10 INTRODUCTION AND PREVIEW
The quantities H, I, C, D, K, W arise naturally in the following areas:
• Data compression. The entropy H of a random variable is a lower bound on the average length of the shortest description of the random variable. We can construct descriptions with average length within 1 bit of the entropy. If we relax the constraint of recovering the source perfectly, we can then ask what communication rates are required to describe the source up to distortion D? And what channel capacities are sufficient to enable the transmission of this source over the channel and its reconstruction with distortion less than or equal to D? This is the subject of rate distortion theory.
When we try to formalize the notion of the shortest description for nonrandom objects, we are led to the definition of Kolmogorov complexity K. Later, we show that Kolmogorov complexity is universal and satisfies many of the intuitive requirements for the theory of shortest descriptions.
• Data transmission. We consider the problem of transmitting information so that the receiver can decode the message with a small probability of error. Essentially, we wish to find codewords (sequences of input symbols to a channel) that are mutually far apart in the sense that their noisy versions (available at the output of the channel) are distinguishable. This is equivalent to sphere packing in highdimensional space. For any set of codewords it is possible to calculate the probability that the receiver will make an error (i.e., make an incorrect decision as to which codeword was sent). However, in most cases, this calculation is tedious.
Using a randomly generated code, Shannon showed that one can send information at any rate below the capacity C of the channel with an arbitrarily low probability of error. The idea of a randomly generated code is very unusual. It provides the basis for a simple analysis of a very difficult problem. One of the key ideas in the proof is the concept of typical sequences. The capacity C is the logarithm of the number of distinguishable input signals.
• Network information theory. Each of the topics mentioned previously involves a single source or a single channel. What if one wishes to compress each of many sources and then put the compressed descriptions together into a joint reconstruction of the sources? This problem is solved by the Slepian–Wolf theorem. Or what if one has many senders sending information independently to a common receiver? What is the channel capacity of this channel? This is the multiple-access channel solved by Liao and Ahlswede. Or what if one has one sender and many

1.1 PREVIEW OF THE BOOK 11
receivers and wishes to communicate (perhaps different) information simultaneously to each of the receivers? This is the broadcast channel. Finally, what if one has an arbitrary number of senders and receivers in an environment of interference and noise. What is the capacity region of achievable rates from the various senders to the receivers? This is the general network information theory problem. All of the preceding problems fall into the general area of multiple-user or network information theory. Although hopes for a comprehensive theory for networks may be beyond current research techniques, there is still some hope that all the answers involve only elaborate forms of mutual information and relative entropy.
• Ergodic theory. The asymptotic equipartition theorem states that most sample n-sequences of an ergodic process have probability about 2−nH and that there are about 2nH such typical sequences.
• Hypothesis testing. The relative entropy D arises as the exponent in the probability of error in a hypothesis test between two distributions. It is a natural measure of distance between distributions.
• Statistical mechanics. The entropy H arises in statistical mechanics as a measure of uncertainty or disorganization in a physical system. Roughly speaking, the entropy is the logarithm of the number of ways in which the physical system can be configured. The second law of thermodynamics says that the entropy of a closed system cannot decrease. Later we provide some interpretations of the second law.
• Quantum mechanics. Here, von Neumann entropy S = tr(ρ ln ρ) = i λi log λi plays the role of classical Shannon–Boltzmann entropy
H = − i pi log pi. Quantum mechanical versions of data compression and channel capacity can then be found.
• Inference. We can use the notion of Kolmogorov complexity K to find the shortest description of the data and use that as a model to predict what comes next. A model that maximizes the uncertainty or entropy yields the maximum entropy approach to inference.
• Gambling and investment. The optimal exponent in the growth rate of wealth is given by the doubling rate W . For a horse race with uniform odds, the sum of the doubling rate W and the entropy H is constant. The increase in the doubling rate due to side information is equal to the mutual information I between a horse race and the side information. Similar results hold for investment in the stock market.
• Probability theory. The asymptotic equipartition property (AEP) shows that most sequences are typical in that they have a sample entropy close to H . So attention can be restricted to these approximately 2nH typical sequences. In large deviation theory, the

12 INTRODUCTION AND PREVIEW
probability of a set is approximately 2−nD, where D is the relative entropy distance between the closest element in the set and the true distribution. • Complexity theory. The Kolmogorov complexity K is a measure of the descriptive complexity of an object. It is related to, but different from, computational complexity, which measures the time or space required for a computation.
Information-theoretic quantities such as entropy and relative entropy arise again and again as the answers to the fundamental questions in communication and statistics. Before studying these questions, we shall study some of the properties of the answers. We begin in Chapter 2 with the definitions and basic properties of entropy, relative entropy, and mutual information.

CHAPTER 2
ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION
In this chapter we introduce most of the basic definitions required for subsequent development of the theory. It is irresistible to play with their relationships and interpretations, taking faith in their later utility. After defining entropy and mutual information, we establish chain rules, the nonnegativity of mutual information, the data-processing inequality, and illustrate these definitions by examining sufficient statistics and Fano’s inequality.
The concept of information is too broad to be captured completely by a single definition. However, for any probability distribution, we define a quantity called the entropy, which has many properties that agree with the intuitive notion of what a measure of information should be. This notion is extended to define mutual information, which is a measure of the amount of information one random variable contains about another. Entropy then becomes the self-information of a random variable. Mutual information is a special case of a more general quantity called relative entropy, which is a measure of the distance between two probability distributions. All these quantities are closely related and share a number of simple properties, some of which we derive in this chapter.
In later chapters we show how these quantities arise as natural answers to a number of questions in communication, statistics, complexity, and gambling. That will be the ultimate test of the value of these definitions.
2.1 ENTROPY
We first introduce the concept of entropy, which is a measure of the uncertainty of a random variable. Let X be a discrete random variable with alphabet X and probability mass function p(x) = Pr{X = x}, x ∈ X.
Elements of Information Theory, Second Edition, By Thomas M. Cover and Joy A. Thomas Copyright  2006 John Wiley & Sons, Inc.
13

14 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

We denote the probability mass function by p(x) rather than pX(x), for convenience. Thus, p(x) and p(y) refer to two different random variables and are in fact different probability mass functions, pX(x) and pY (y), respectively.
Definition The entropy H (X) of a discrete random variable X is defined by

H (X) = − p(x) log p(x).

(2.1)

x∈X

We also write H (p) for the above quantity. The log is to the base 2

and entropy is expressed in bits. For example, the entropy of a fair coin toss is 1 bit. We will use the convention that 0 log 0 = 0, which is easily justified by continuity since x log x → 0 as x → 0. Adding terms of zero

probability does not change the entropy.
If the base of the logarithm is b, we denote the entropy as Hb(X). If the base of the logarithm is e, the entropy is measured in nats. Unless

otherwise specified, we will take all logarithms to base 2, and hence all

the entropies will be measured in bits. Note that entropy is a functional of the distribution of X. It does not depend on the actual values taken by the random variable X, but only on the probabilities.
We denote expectation by E. Thus, if X ∼ p(x), the expected value of the random variable g(X) is written

Epg(X) = g(x)p(x),
x∈X

(2.2)

or more simply as Eg(X) when the probability mass function is under-

stood from the context. We shall take a peculiar interest in the eerily

self-referential

expectation

of

g(X)

under

p(x)

when

g(X)

=

log

1 p(X)

.

Remark The entropy of X can also be interpreted as the expected value

of

the

random

variable

log

1 p(X)

,

where

X

is

drawn

according

to

probability

mass function p(x). Thus,

1

H (X) = Ep log p(X).

(2.3)

This definition of entropy is related to the definition of entropy in thermodynamics; some of the connections are explored later. It is possible to derive the definition of entropy axiomatically by defining certain properties that the entropy of a random variable must satisfy. This approach is illustrated in Problem 2.46. We do not use the axiomatic approach to

2.1 ENTROPY 15

justify the definition of entropy; instead, we show that it arises as the answer to a number of natural questions, such as “What is the average length of the shortest description of the random variable?” First, we derive some immediate consequences of the definition.

Lemma 2.1.1 H (X) ≥ 0.

Proof:

0

≤

p(x)

≤

1

implies

that

log

1 p(x)

≥

0.

Lemma 2.1.2 Hb(X) = (logb a)Ha(X).

Proof: logb p = logb a loga p.

The second property of entropy enables us to change the base of the logarithm in the definition. Entropy can be changed from one base to another by multiplying by the appropriate factor.

Example 2.1.1 Let

X=

1 0

with probability p, with probability 1 − p.

(2.4)

Then

H (X) = −p log p − (1 − p) log(1 − p) =de=f H (p).

(2.5)

In

particular,

H (X)

=

1

bit

when

p

=

1 2

.

The

graph

of

the

function

H (p)

is shown in Figure 2.1. The figure illustrates some of the basic properties

of entropy: It is a concave function of the distribution and equals 0 when

p = 0 or 1. This makes sense, because when p = 0 or 1, the variable

is not random and there is no uncertainty. Similarly, the uncertainty is

maximum

when

p

=

1 2

,

which

also

corresponds

to

the

maximum

value

of

the entropy.

Example 2.1.2 Let



X

 = 

a b c d

with

probability

1 2

,

with

probability

1 4

,

with

probability

1 8

,

with

probability

1 8

.

(2.6)

The entropy of X is

H (X) = − 1

log

1

−

1

log

1

−

11 log

−

11 log

=

7

bits.

(2.7)

2 24 48 88 8 4

H(p)

16 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p
FIGURE 2.1. H (p) vs. p.
Suppose that we wish to determine the value of X with the minimum number of binary questions. An efficient first question is “Is X = a?” This splits the probability in half. If the answer to the first question is no, the second question can be “Is X = b?” The third question can be “Is X = c?” The resulting expected number of binary questions required is 1.75. This turns out to be the minimum expected number of binary questions required to determine the value of X. In Chapter 5 we show that the minimum expected number of binary questions required to determine X lies between H (X) and H (X) + 1.

2.2 JOINT ENTROPY AND CONDITIONAL ENTROPY

We defined the entropy of a single random variable in Section 2.1. We now extend the definition to a pair of random variables. There is nothing really new in this definition because (X, Y ) can be considered to be a single vector-valued random variable.

Definition The joint entropy H (X, Y ) of a pair of discrete random variables (X, Y ) with a joint distribution p(x, y) is defined as

H (X, Y ) = −

p(x, y) log p(x, y),

x∈X y∈Y

(2.8)

2.2 JOINT ENTROPY AND CONDITIONAL ENTROPY

17

which can also be expressed as

H (X, Y ) = −E log p(X, Y ).

(2.9)

We also define the conditional entropy of a random variable given another as the expected value of the entropies of the conditional distributions, averaged over the conditioning random variable.

Definition If (X, Y ) ∼ p(x, y), the conditional entropy H (Y |X) is defined as

H (Y |X) = p(x)H (Y |X = x)
x∈X

= − p(x) p(y|x) log p(y|x)

x∈X

y∈Y

(2.10) (2.11)

=−

p(x, y) log p(y|x)

x∈X y∈Y

= −E log p(Y |X).

(2.12) (2.13)

The naturalness of the definition of joint entropy and conditional entropy is exhibited by the fact that the entropy of a pair of random variables is the entropy of one plus the conditional entropy of the other. This is proved in the following theorem.

Theorem 2.2.1 (Chain rule)

H (X, Y ) = H (X) + H (Y |X).

(2.14)

Proof

H (X, Y ) = −

p(x, y) log p(x, y)

x∈X y∈Y

(2.15)

=−

p(x, y) log p(x)p(y|x)

x∈X y∈Y

(2.16)

=−

p(x, y) log p(x) −

p(x, y) log p(y|x)

x∈X y∈Y

x∈X y∈Y

(2.17)

= − p(x) log p(x) −

p(x, y) log p(y|x)

x∈X

x∈X y∈Y

= H (X) + H (Y |X).

(2.18) (2.19)

18 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Equivalently, we can write log p(X, Y ) = log p(X) + log p(Y |X)

(2.20)

and take the expectation of both sides of the equation to obtain the theorem.

Corollary

H (X, Y |Z) = H (X|Z) + H (Y |X, Z).

(2.21)

Proof: The proof follows along the same lines as the theorem.

Example 2.2.1 Let (X, Y ) have the following joint distribution:

X

y

1

2

3

4

I

I

I

1

8

16

32

32

2

I
16

I
8

I
32

I
32

3

I 16

J
16

l
16

I
16

4

I
4

0

0

0

The

marginal

distribution

of

X

is

(

1 2

,

1 4

,

1 8

,

1 8

)

and

the

marginal

distribution

of

Y

is

(

1 4

,

1 4

,

1 4

,

1 4

),

and

hence

H (X) =

7 4

bits

and

H (Y )

=2

bits.

Also,

4
H (X|Y ) = p(Y = i)H (X|Y = i)

i=1

=

1 H

1111 ,,,

+

1 H

1111 ,,,

4 2488 4 4288

(2.22)

+ 1 H 1 , 1 , 1 , 1 + 1 H (1, 0, 0, 0) 4 4444 4

= 1 × 7 + 1 × 7 + 1 ×2+ 1 ×0

44444

4

= 11 bits. 8

(2.23) (2.24) (2.25)

Similarly,

H (Y |X)

=

13 8

bits

and

H (X, Y )

=

27 8

bits.

Remark Note that H (Y |X) = H (X|Y ). However, H (X) − H (X|Y ) = H (Y )− H (Y |X), a property that we exploit later.

2.3 RELATIVE ENTROPY AND MUTUAL INFORMATION 19
2.3 RELATIVE ENTROPY AND MUTUAL INFORMATION

The entropy of a random variable is a measure of the uncertainty of the random variable; it is a measure of the amount of information required on the average to describe the random variable. In this section we introduce two related concepts: relative entropy and mutual information.
The relative entropy is a measure of the distance between two distributions. In statistics, it arises as an expected logarithm of the likelihood ratio. The relative entropy D(p||q) is a measure of the inefficiency of assuming that the distribution is q when the true distribution is p. For example, if we knew the true distribution p of the random variable, we could construct a code with average description length H (p). If, instead, we used the code for a distribution q, we would need H (p) + D(p||q) bits on the average to describe the random variable.

Definition The relative entropy or Kullback–Leibler distance between two probability mass functions p(x) and q(x) is defined as

p(x) D(p||q) = p(x) log
q(x)
x∈X

(2.26)

p(X) = Ep log q(X) .

(2.27)

In the above definition, we use the convention convention (based on continuity arguments) that 0

that log

0 log

0 q

=

0

0 0

=

and

0 p

and the

log

p 0

=

∞. Thus, if there is any symbol x ∈ X such that p(x) > 0 and q(x) = 0,

then D(p||q) = ∞.

We will soon show that relative entropy is always nonnegative and is

zero if and only if p = q. However, it is not a true distance between

distributions since it is not symmetric and does not satisfy the triangle

inequality. Nonetheless, it is often useful to think of relative entropy as a

“distance” between distributions.

We now introduce mutual information, which is a measure of the

amount of information that one random variable contains about another

random variable. It is the reduction in the uncertainty of one random

variable due to the knowledge of the other.

Definition Consider two random variables X and Y with a joint probability mass function p(x, y) and marginal probability mass functions p(x) and p(y). The mutual information I (X; Y ) is the relative entropy between

20 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

the joint distribution and the product distribution p(x)p(y):

p(x, y)

I (X; Y ) =

p(x, y) log

x∈X y∈Y

p(x)p(y)

(2.28)

= D(p(x, y)||p(x)p(y))

(2.29)

p(X, Y ) = Ep(x,y) log p(X)p(Y ) .

(2.30)

In Chapter 8 we generalize this definition to continuous random variables, and in (8.54) to general random variables that could be a mixture of discrete and continuous random variables.

Example 2.3.1 Let X = {0, 1} and consider two distributions p and q on X. Let p(0) = 1 − r, p(1) = r, and let q(0) = 1 − s, q(1) = s. Then

D(p||q)

=

(1 − r) log

1 1

− −

r s

+

r

log

r s

(2.31)

and

D(q||p)

=

(1

−

s) log

1−s 1−r

+ s log

s .
r

(2.32)

If

r

=

s,

then

D(p||q)

=

D(q||p)

=

0.

If

r

=

1 2

,

s

=

1 4

,

we

can

calculate

D(p||q)

=

1 log
2

1 2
3 4

+

1 2

log

1 2
1 4

=1−

1 log 3
2

=

0.2075

bit,

(2.33)

whereas

D(q||p)

=

3 log
4

3 4
1 2

+

1 log
4

1 4
1 2

=

3 4

log 3 − 1

=

0.1887

bit.

(2.34)

Note that D(p||q) = D(q||p) in general.

2.4 RELATIONSHIP BETWEEN ENTROPY AND MUTUAL INFORMATION

We can rewrite the definition of mutual information I (X; Y ) as

p(x, y)

I (X; Y ) = p(x, y) log

x,y

p(x)p(y)

(2.35)

2.4 RELATIONSHIP BETWEEN ENTROPY AND MUTUAL INFORMATION 21

p(x|y)

= p(x, y) log

x,y

p(x)

= − p(x, y) log p(x) + p(x, y) log p(x|y)

x,y

x,y

(2.36) (2.37)

= − p(x) log p(x) − − p(x, y) log p(x|y) (2.38)

x

x,y

= H (X) − H (X|Y ).

(2.39)

Thus, the mutual information I (X; Y ) is the reduction in the uncertainty of X due to the knowledge of Y .
By symmetry, it also follows that

I (X; Y ) = H (Y ) − H (Y |X).

(2.40)

Thus, X says as much about Y as Y says about X. Since H (X, Y ) = H (X) + H (Y |X), as shown in Section 2.2, we have

I (X; Y ) = H (X) + H (Y ) − H (X, Y ).

(2.41)

Finally, we note that

I (X; X) = H (X) − H (X|X) = H (X).

(2.42)

Thus, the mutual information of a random variable with itself is the entropy of the random variable. This is the reason that entropy is sometimes referred to as self-information.
Collecting these results, we have the following theorem.

Theorem 2.4.1 (Mutual information and entropy)

I (X; Y ) = H (X) − H (X|Y ) I (X; Y ) = H (Y ) − H (Y |X) I (X; Y ) = H (X) + H (Y ) − H (X, Y ) I (X; Y ) = I (Y ; X) I (X; X) = H (X).

(2.43) (2.44) (2.45) (2.46) (2.47)

22 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION
H(X,Y )

H(X|Y ) I(X;Y ) H(Y |X )

H(X )

H(Y )

FIGURE 2.2. Relationship between entropy and mutual information.

The relationship between H (X), H (Y ), H (X, Y ), H (X|Y ), H (Y |X), and I (X; Y ) is expressed in a Venn diagram (Figure 2.2). Notice that the mutual information I (X; Y ) corresponds to the intersection of the information in X with the information in Y .

Example 2.4.1 For the joint distribution of Example 2.2.1, it is easy to calculate the mutual information I (X; Y ) = H (X) − H (X|Y ) = H (Y ) − H (Y |X) = 0.375 bit.

2.5 CHAIN RULES FOR ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

We now show that the entropy of a collection of random variables is the sum of the conditional entropies.

Theorem 2.5.1 (Chain rule for entropy) Let X1, X2, . . . , Xn be drawn according to p(x1, x2, . . . , xn). Then

n
H (X1, X2, . . . , Xn) = H (Xi|Xi−1, . . . , X1).
i=1

(2.48)

Proof: By repeated application of the two-variable expansion rule for entropies, we have

H (X1, X2) = H (X1) + H (X2|X1), H (X1, X2, X3) = H (X1) + H (X2, X3|X1)

(2.49) (2.50)

2.5 CHAIN RULES FOR ENTROPY, RELATIVE ENTROPY, MUTUAL INFORMATION 23

= H (X1) + H (X2|X1) + H (X3|X2, X1), ...

(2.51)

H (X1, X2, . . . , Xn) = H (X1) + H (X2|X1) + · · · + H (Xn|Xn−1, . . . , X1)

n
= H (Xi|Xi−1, . . . , X1).
i=1
Alternative Proof: We write p(x1, . . . , xn) = and evaluate

(2.52) (2.53)

n i=1

p(xi

|xi−1,

.

.

.

,

x1)

H (X1, X2, . . . , Xn)

=−

p(x1, x2, . . . , xn) log p(x1, x2, . . . , xn)

x1,x2,...,xn

n

=−

p(x1, x2, . . . , xn) log p(xi|xi−1, . . . , x1)

x1,x2,...,xn

i=1

n

=−

p(x1, x2, . . . , xn) log p(xi|xi−1, . . . , x1)

x1,x2,...,xn i=1 n

=−

p(x1, x2, . . . , xn) log p(xi|xi−1, . . . , x1)

i=1 x1,x2,...,xn n

=−

p(x1, x2, . . . , xi) log p(xi|xi−1, . . . , x1)

i=1 x1,x2,...,xi n

= H (Xi|Xi−1, . . . , X1).

i=1

(2.54) (2.55) (2.56) (2.57) (2.58) (2.59)

We now define the conditional mutual information as the reduction in the uncertainty of X due to knowledge of Y when Z is given.

Definition The conditional mutual information of random variables X and Y given Z is defined by

I (X; Y |Z) = H (X|Z) − H (X|Y, Z)

=

Ep(x,y,z)

log

p(X, Y |Z) p(X|Z)p(Y |Z)

.

(2.60) (2.61)

Mutual information also satisfies a chain rule.

24 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Theorem 2.5.2 (Chain rule for information)
n
I (X1, X2, . . . , Xn; Y ) = I (Xi; Y |Xi−1, Xi−2, . . . , X1).
i=1
Proof

(2.62)

I (X1, X2, . . . , Xn; Y )

= H (X1, X2, . . . , Xn) − H (X1, X2, . . . , Xn|Y )

(2.63)

n

n

= H (Xi|Xi−1, . . . , X1) − H (Xi|Xi−1, . . . , X1, Y )

i=1

i=1

n
= I (Xi; Y |X1, X2, . . . , Xi−1).
i=1

(2.64)

We define a conditional version of the relative entropy.

Definition For joint probability mass functions p(x, y) and q(x, y), the conditional relative entropy D(p(y|x)||q(y|x)) is the average of the relative entropies between the conditional probability mass functions p(y|x) and q(y|x) averaged over the probability mass function p(x). More pre-
cisely,

D(p(y|x)||q(y|x)) = p(x) p(y|x) log p(y|x)

x

y

q(y|x)

=

Ep(x,y)

log

p(Y q(Y

|X) |X)

.

(2.65) (2.66)

The notation for conditional relative entropy is not explicit since it omits mention of the distribution p(x) of the conditioning random variable. However, it is normally understood from the context.
The relative entropy between two joint distributions on a pair of random variables can be expanded as the sum of a relative entropy and a conditional relative entropy. The chain rule for relative entropy is used in Section 4.4 to prove a version of the second law of thermodynamics.

Theorem 2.5.3 (Chain rule for relative entropy)

D(p(x, y)||q(x, y)) = D(p(x)||q(x)) + D(p(y|x)||q(y|x)). (2.67)

2.6 JENSEN’S INEQUALITY AND ITS CONSEQUENCES 25

Proof

D(p(x, y)||q(x, y))

=

p(x, y) p(x, y) log

xy

q(x, y)

=

p(x)p(y|x) p(x, y) log

xy

q(x)q(y|x)

=
x

p(x, y) log p(x) +

y

q(x) x

p(y|x) y p(x, y) log q(y|x)

= D(p(x)||q(x)) + D(p(y|x)||q(y|x)).

(2.68) (2.69) (2.70) (2.71)

2.6 JENSEN’S INEQUALITY AND ITS CONSEQUENCES

In this section we prove some simple properties of the quantities defined earlier. We begin with the properties of convex functions.

Definition A function f (x) is said to be convex over an interval (a, b) if for every x1, x2 ∈ (a, b) and 0 ≤ λ ≤ 1,

f (λx1 + (1 − λ)x2) ≤ λf (x1) + (1 − λ)f (x2).

(2.72)

A function f is said to be strictly convex if equality holds only if λ = 0 or λ = 1.

Definition A function f is concave if −f is convex. A function is convex if it always lies below any chord. A function is concave if it
always lies above any chord. Examples of convex functions include x2, |x|, ex, x log x (f√or x ≥
0), and so on. Examples of concave functions include log x and x for x ≥ 0. Figure 2.3 shows some examples of convex and concave functions. Note that linear functions ax + b are both convex and concave. Convexity underlies many of the basic properties of information-theoretic quantities
such as entropy and mutual information. Before we prove some of these
properties, we derive some simple results for convex functions.

Theorem 2.6.1 If the function f has a second derivative that is nonnegative (positive) over an interval, the function is convex (strictly convex) over that interval.

26 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

(a)

(b) FIGURE 2.3. Examples of (a) convex and (b) concave functions.

Proof: We use the Taylor series expansion of the function around x0:

f (x) = f (x0) + f

(x0)(x − x0) + f

(x∗) 2

(x

−

x0)2,

(2.73)

where x∗ lies between x0 and x. By hypothesis, f (x∗) ≥ 0, and thus the last term is nonnegative for all x.
We let x0 = λx1 + (1 − λ)x2 and take x = x1, to obtain

f (x1) ≥ f (x0) + f (x0)((1 − λ)(x1 − x2)).

(2.74)

Similarly, taking x = x2, we obtain f (x2) ≥ f (x0) + f (x0)(λ(x2 − x1)).

(2.75)

Multiplying (2.74) by λ and (2.75) by 1 − λ and adding, we obtain (2.72). The proof for strict convexity proceeds along the same lines.

x2

Theorem 2.6.1 allows , ex, and x log x for x

us immediately to verify the strict conve√xity of ≥ 0, and the strict concavity of log x and x for

x ≥ 0.

Let E denote expectation. Thus, EX = x∈X p(x)x in the discrete case and EX = xf (x) dx in the continuous case.

2.6 JENSEN’S INEQUALITY AND ITS CONSEQUENCES 27

The next inequality is one of the most widely used in mathematics and one that underlies many of the basic results in information theory.

Theorem 2.6.2 (Jensen’s inequality) If f is a convex function and X is a random variable,

Ef (X) ≥ f (EX).

(2.76)

Moreover, if f is strictly convex, the equality in (2.76) implies that X = EX with probability 1 (i.e., X is a constant).

Proof: We prove this for discrete distributions by induction on the number of mass points. The proof of conditions for equality when f is strictly convex is left to the reader.
For a two-mass-point distribution, the inequality becomes

p1f (x1) + p2f (x2) ≥ f (p1x1 + p2x2),

(2.77)

which follows directly from the definition of convex functions. Suppose that the theorem is true for distributions with k − 1 mass points. Then writing pi = pi/(1 − pk) for i = 1, 2, . . . , k − 1, we have

k

k−1

pif (xi) = pkf (xk) + (1 − pk) pif (xi)

i=1

i=1

≥ pkf (xk) + (1 − pk)f

k−1
pi xi
i=1

k−1
≥ f pkxk + (1 − pk) pixi
i=1

k

=f

pixi ,

i=1

(2.78) (2.79) (2.80) (2.81)

where the first inequality follows from the induction hypothesis and the second follows from the definition of convexity.
The proof can be extended to continuous distributions by continuity arguments.

We now use these results to prove some of the properties of entropy and relative entropy. The following theorem is of fundamental importance.

28 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION
Theorem 2.6.3 (Information inequality) Let p(x), q(x), x ∈ X, be two probability mass functions. Then

D(p||q) ≥ 0

(2.82)

with equality if and only if p(x) = q(x) for all x.

Proof: Let A = {x : p(x) > 0} be the support set of p(x). Then

−D(p||q) = − p(x) log p(x)

x∈A

q(x)

q(x) = p(x) log
p(x)
x∈A

≤ log p(x) q(x)

x∈A

p(x)

= log q(x)
x∈A

≤ log q(x)
x∈X
= log 1

= 0,

(2.83)
(2.84)
(2.85) (2.86) (2.87) (2.88) (2.89)

where (2.85) follows from Jensen’s inequality. Since log t is a strictly concave function of t, we have equality in (2.85) if and only if q(x)/p(x) is constant everywhere [i.e., q(x) = cp(x) for all x]. Thus, x∈A q(x) = c x∈A p(x) = c. We have equality in (2.87) only if x∈A q(x) = x∈X q(x) = 1, which implies that c = 1. Hence, we have D(p||q) = 0 if and only if p(x) = q(x) for all x.

Corollary (Nonnegativity of mutual information) For any two random

variables, X, Y ,

I (X; Y ) ≥ 0,

(2.90)

with equality if and only if X and Y are independent.

Proof: I (X; Y ) = D(p(x, y)||p(x)p(y)) ≥ 0, with equality if and only if p(x, y) = p(x)p(y) (i.e., X and Y are independent).

2.6 JENSEN’S INEQUALITY AND ITS CONSEQUENCES 29

Corollary

D(p(y|x)||q(y|x)) ≥ 0,

(2.91)

with equality if and only if p(y|x) = q(y|x) for all y and x such that p(x) > 0.

Corollary

I (X; Y |Z) ≥ 0,

(2.92)

with equality if and only if X and Y are conditionally independent given Z.

We now show that the uniform distribution over the range X is the maximum entropy distribution over this range. It follows that any random variable with this range has an entropy no greater than log |X|.

Theorem 2.6.4 H (X) ≤ log |X|, where |X| denotes the number of elements in the range of X, with equality if and only X has a uniform distribution over X.

Proof:

Let

u(x)

=

1 |X |

be

the

uniform

probability

mass

function

over

X,

and let p(x) be the probability mass function for X. Then

D(p u) = p(x) log p(x) = log |X| − H (X). u(x)

(2.93)

Hence by the nonnegativity of relative entropy,

0 ≤ D(p u) = log |X| − H (X).

(2.94)

Theorem 2.6.5 (Conditioning reduces entropy)(Information can’t hurt)

H (X|Y ) ≤ H (X)

(2.95)

with equality if and only if X and Y are independent.

Proof: 0 ≤ I (X; Y ) = H (X) − H (X|Y ).

Intuitively, the theorem says that knowing another random variable Y can only reduce the uncertainty in X. Note that this is true only on the average. Specifically, H (X|Y = y) may be greater than or less than or equal to H (X), but on the average H (X|Y ) = y p(y)H (X|Y = y) ≤ H (X). For example, in a court case, specific new evidence might increase
uncertainty, but on the average evidence decreases uncertainty.

30 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Example 2.6.1 Let (X, Y ) have the following joint distribution:

X

y

2

0

1
4

2

I
8

I
8

Then

H

(X)

=

H

(

1 8

,

7 8

)

=

0.544

bit,

H (X|Y = 1) = 0

bits,

and

H (X|Y = 2) = 1

bit.

We

calculate

H (X|Y )

=

3 4

H

(X|Y

=

1)

+

1 4

H (X|Y = 2) = 0.25 bit. Thus, the uncertainty in X is increased if Y = 2

is observed and decreased if Y = 1 is observed, but uncertainty decreases

on the average.

Theorem 2.6.6 (Independence bound on entropy) X1, X2, . . . , Xn be drawn according to p(x1, x2, . . . , xn). Then
n
H (X1, X2, . . . , Xn) ≤ H (Xi)
i=1

Let (2.96)

with equality if and only if the Xi are independent.

Proof: By the chain rule for entropies,

n
H (X1, X2, . . . , Xn) = H (Xi|Xi−1, . . . , X1)
i=1
n
≤ H (Xi),
i=1

(2.97) (2.98)

where the inequality follows directly from Theorem 2.6.5. We have equal-
ity if and only if Xi is independent of Xi−1, . . . , X1 for all i (i.e., if and only if the Xi’s are independent).

2.7 LOG SUM INEQUALITY AND ITS APPLICATIONS
We now prove a simple consequence of the concavity of the logarithm, which will be used to prove some concavity results for the entropy.

2.7 LOG SUM INEQUALITY AND ITS APPLICATIONS 31

Theorem 2.7.1 (Log sum inequality) For nonnegative numbers, a1, a2, . . . , an and b1, b2, . . . , bn,

n i=1

ai

log

ai bi

≥

n
ai
i=1

log

n i=1

ai

n i=1

bi

(2.99)

with

equality

if

and

only

if

ai bi

=

const.

We

again

use

the

convention

that

0 log 0

=

0,

a

log

a 0

=

∞

if

a

>

0

and

0

log

0 0

=

0.

These

follow

easily

from

continuity.

Proof: Assume without loss of generality that ai > 0 and bi > 0. The

function f (t) = t log t is strictly convex, since f

(t) =

1 t

log

e

>

0

for

all

positive t. Hence by Jensen’s inequality, we have

αif (ti) ≥ f

αi ti

(2.100)

for αi ≥ 0,

i αi = 1. Setting αi =

bi

n j =1

bj

and ti =

ai bi

,

we

obtain

ai log ai ≥ bj bi

ai log bj

ai , bj

(2.101)

which is the log sum inequality.

We now use the log sum inequality to prove various convexity results. We begin by reproving Theorem 2.6.3, which states that D(p||q) ≥ 0 with equality if and only if p(x) = q(x). By the log sum inequality,

D(p||q) =

p(x

)

log

p(x) q(x)

(2.102)

≥

p(x) log p(x)

q(x)

(2.103)

= 1 log 1 = 0 1

(2.104)

with

equality

if

and

only

if

p(x) q(x)

=

c.

Since

both

p

and

q

are

probability

mass functions, c = 1, and hence we have D(p||q) = 0 if and only if

p(x) = q(x) for all x.

32 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Theorem 2.7.2 (Convexity of relative entropy) D(p||q) is convex in the pair (p, q); that is, if (p1, q1) and (p2, q2) are two pairs of probability mass functions, then

D(λp1 + (1 − λ)p2||λq1 + (1 − λ)q2) ≤ λD(p1||q1) + (1 − λ)D(p2||q2) (2.105)
for all 0 ≤ λ ≤ 1.

Proof: We apply the log sum inequality to a term on the left-hand side of (2.105):

(λp1(x)

+

(1

−

λ)p2(x))

log

λp1(x) λq1(x)

+ +

(1 (1

− −

λ)p2(x) λ)q2(x)

≤

λp1(x)

log

λp1(x) λq1(x)

+

(1

−

λ)p2(x) log

(1 − (1 −

λ)p2(x) . λ)q2(x)

(2.106)

Summing this over all x, we obtain the desired property.

Theorem 2.7.3 (Concavity of entropy) H (p) is a concave function of p.

Proof

H (p) = log |X| − D(p||u),

(2.107)

where u is the uniform distribution on |X| outcomes. The concavity of H then follows directly from the convexity of D.

Alternative Proof: Let X1 be a random variable with distribution p1, taking on values in a set A. Let X2 be another random variable with distribution p2 on the same set. Let

θ=

1 2

with probability λ, with probability 1 − λ.

(2.108)

Let Z = Xθ . Then the distribution of Z is λp1 + (1 − λ)p2. Now since conditioning reduces entropy, we have

H (Z) ≥ H (Z|θ ),

(2.109)

or equivalently,

H (λp1 + (1 − λ)p2) ≥ λH (p1) + (1 − λ)H (p2),

(2.110)

which proves the concavity of the entropy as a function of the distribution.

2.7 LOG SUM INEQUALITY AND ITS APPLICATIONS 33

One of the consequences of the concavity of entropy is that mixing two gases of equal entropy results in a gas with higher entropy.

Theorem 2.7.4 Let (X, Y ) ∼ p(x, y) = p(x)p(y|x). The mutual information I (X; Y ) is a concave function of p(x) for fixed p(y|x) and a convex function of p(y|x) for fixed p(x).

Proof: To prove the first part, we expand the mutual information

I (X; Y ) = H (Y ) − H (Y |X) = H (Y ) − p(x)H (Y |X = x). (2.111)
x
If p(y|x) is fixed, then p(y) is a linear function of p(x). Hence H (Y ), which is a concave function of p(y), is a concave function of p(x). The second term is a linear function of p(x). Hence, the difference is a concave function of p(x).
To prove the second part, we fix p(x) and consider two different conditional distributions p1(y|x) and p2(y|x). The corresponding joint distributions are p1(x, y) = p(x)p1(y|x) and p2(x, y) = p(x)p2(y|x), and their respective marginals are p(x), p1(y) and p(x), p2(y). Consider a conditional distribution

pλ(y|x) = λp1(y|x) + (1 − λ)p2(y|x),

(2.112)

which is a mixture of p1(y|x) and p2(y|x) where 0 ≤ λ ≤ 1. The corresponding joint distribution is also a mixture of the corresponding joint distributions,

pλ(x, y) = λp1(x, y) + (1 − λ)p2(x, y), and the distribution of Y is also a mixture,

(2.113)

pλ(y) = λp1(y) + (1 − λ)p2(y).

(2.114)

Hence if we let qλ(x, y) = p(x)pλ(y) be the product of the marginal distributions, we have

qλ(x, y) = λq1(x, y) + (1 − λ)q2(x, y).

(2.115)

Since the mutual information is the relative entropy between the joint distribution and the product of the marginals,

I (X; Y ) = D(pλ(x, y)||qλ(x, y)),

(2.116)

and relative entropy D(p||q) is a convex function of (p, q), it follows that the mutual information is a convex function of the conditional distribution.

34 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

2.8 DATA-PROCESSING INEQUALITY

The data-processing inequality can be used to show that no clever manipulation of the data can improve the inferences that can be made from the data.

Definition Random variables X, Y, Z are said to form a Markov chain in that order (denoted by X → Y → Z) if the conditional distribution of Z depends only on Y and is conditionally independent of X. Specifically, X, Y , and Z form a Markov chain X → Y → Z if the joint probability
mass function can be written as

p(x, y, z) = p(x)p(y|x)p(z|y).

(2.117)

Some simple consequences are as follows:

• X → Y → Z if and only if X and Z are conditionally independent given Y . Markovity implies conditional independence because

p(x, z|y) = p(x, y, z) = p(x, y)p(z|y) = p(x|y)p(z|y). (2.118)

p(y)

p(y)

This is the characterization of Markov chains that can be extended to define Markov fields, which are n-dimensional random processes in which the interior and exterior are independent given the values on the boundary.
• X → Y → Z implies that Z → Y → X. Thus, the condition is sometimes written X ↔ Y ↔ Z.
• If Z = f (Y ), then X → Y → Z.

We can now prove an important and useful theorem demonstrating that no processing of Y , deterministic or random, can increase the information that Y contains about X.

Theorem 2.8.1 (Data-processing inequality) I (X; Y ) ≥ I (X; Z).

If X → Y → Z, then

Proof: By the chain rule, we can expand mutual information in two different ways:

I (X; Y, Z) = I (X; Z) + I (X; Y |Z) = I (X; Y ) + I (X; Z|Y ).

(2.119) (2.120)

2.9 SUFFICIENT STATISTICS 35

Since X and Z are conditionally independent given Y , we have I (X; Z|Y ) = 0. Since I (X; Y |Z) ≥ 0, we have

I (X; Y ) ≥ I (X; Z).

(2.121)

We have equality if and only if I (X; Y |Z) = 0 (i.e., X → Z → Y forms a Markov chain). Similarly, one can prove that I (Y ; Z) ≥ I (X; Z).

Corollary In particular, if Z = g(Y ), we have I (X; Y ) ≥ I (X; g(Y )).

Proof: X → Y → g(Y ) forms a Markov chain.

Thus functions of the data Y cannot increase the information about X.

Corollary If X → Y → Z, then I (X; Y |Z) ≤ I (X; Y ).

Proof: We note in (2.119) and (2.120) that I (X; Z|Y ) = 0, by Markovity, and I (X; Z) ≥ 0. Thus,

I (X; Y |Z) ≤ I (X; Y ).

(2.122)

Thus, the dependence of X and Y is decreased (or remains unchanged)

by the observation of a “downstream” random variable Z. Note that it is

also possible that I (X; Y |Z) > I (X; Y ) when X, Y , and Z do not form a

Markov chain. For example, let X and Y be independent fair binary ran-

dom variables, and let Z = X + Y . Then I (X; Y ) = 0, but I (X; Y |Z) =

H (X|Z) − H (X|Y, Z)

=

H (X|Z)

=

P (Z

=

1)H (X|Z

=

1)

=

-1
2

bit.

2.9 SUFFICIENT STATISTICS

This section is a sidelight showing the power of the data-processing
inequality in clarifying an important idea in statistics. Suppose that we have a family of probability mass functions {fθ (x)} indexed by θ , and let X be a sample from a distribution in this family. Let T (X) be any statistic
(function of the sample) like the sample mean or sample variance. Then θ → X → T (X), and by the data-processing inequality, we have

I (θ; T (X)) ≤ I (θ; X)

(2.123)

for any distribution on θ . However, if equality holds, no information is lost.
A statistic T (X) is called sufficient for θ if it contains all the information in X about θ .

36 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION
Definition A function T (X) is said to be a sufficient statistic relative to the family {fθ (x)} if X is independent of θ given T (X) for any distribution on θ [i.e., θ → T (X) → X forms a Markov chain].
This is the same as the condition for equality in the data-processing inequality,

I (θ; X) = I (θ; T (X))

(2.124)

for all distributions on θ . Hence sufficient statistics preserve mutual information and conversely.
Here are some examples of sufficient statistics:

1. Let X1, X2, . . . , Xn, Xi ∈ {0, 1}, be an independent and identically

distributed (i.i.d.) sequence of coin tosses of a coin with unknown

parameter θ = Pr(Xi = 1). Given n, the number of 1’s is a sufficient

statistic for θ . Here T (X1, X2, . . . , Xn) =

n i=1

Xi .

In

fact,

we

can

show that given T , all sequences having that many 1’s are equally

likely and independent of the parameter θ . Specifically,

n
Pr (X1, X2, . . . , Xn) = (x1, x2, . . . , xn) Xi = k
i=1

=

1 (nk)

if

xi = k,

0 otherwise.

(2.125)

Thus, θ → Xi → (X1, X2, . . . , Xn) forms a Markov chain, and T is a sufficient statistic for θ .
The next two examples involve probability densities instead of probability mass functions, but the theory still applies. We define entropy and mutual information for continuous random variables in Chapter 8.
2. If X is normally distributed with mean θ and variance 1; that is, if

1 fθ (x) = √

e−(x−θ)2/2 = N(θ, 1),

2π

(2.126)

and X1, X2, . . . , Xn are drawn independently according to this distri-

bution,

a

sufficient

statistic

for

θ

is

the

sample

mean

Xn

=

1 n

n i=1

Xi

.

It can be verified that the conditional distribution of X1, X2, . . . , Xn,

conditioned on Xn and n does not depend on θ .

2.10 FANO’S INEQUALITY 37

3. If fθ = Uniform(θ, θ + 1), a sufficient statistic for θ is
T (X1, X2, . . . , Xn) = (max{X1, X2, . . . , Xn}, min{X1, X2, . . . , Xn}).

(2.127)

The proof of this is slightly more complicated, but again one can show that the distribution of the data is independent of the parameter given the statistic T .

The minimal sufficient statistic is a sufficient statistic that is a function of all other sufficient statistics.

Definition A statistic T (X) is a minimal sufficient statistic relative to {fθ (x)} if it is a function of every other sufficient statistic U . Interpreting this in terms of the data-processing inequality, this implies that

θ → T (X) → U (X) → X.

(2.128)

Hence, a minimal sufficient statistic maximally compresses the information about θ in the sample. Other sufficient statistics may contain additional irrelevant information. For example, for a normal distribution with mean θ , the pair of functions giving the mean of all odd samples and the mean of all even samples is a sufficient statistic, but not a minimal sufficient statistic. In the preceding examples, the sufficient statistics are also minimal.

2.10 FANO’S INEQUALITY
Suppose that we know a random variable Y and we wish to guess the value of a correlated random variable X. Fano’s inequality relates the probability of error in guessing the random variable X to its conditional entropy H (X|Y ). It will be crucial in proving the converse to Shannon’s channel capacity theorem in Chapter 7. From Problem 2.5 we know that the conditional entropy of a random variable X given another random variable Y is zero if and only if X is a function of Y . Hence we can estimate X from Y with zero probability of error if and only if H (X|Y ) = 0.
Extending this argument, we expect to be able to estimate X with a low probability of error only if the conditional entropy H (X|Y ) is small. Fano’s inequality quantifies this idea. Suppose that we wish to estimate a random variable X with a distribution p(x). We observe a random variable Y that is related to X by the conditional distribution p(y|x). From Y , we

38 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

calculate a function g(Y ) = Xˆ , where Xˆ is an estimate of X and takes on values in Xˆ . We will not restrict the alphabet Xˆ to be equal to X, and we
will also allow the function g(Y ) to be random. We wish to bound the probability that Xˆ = X. We observe that X → Y → Xˆ forms a Markov
chain. Define the probability of error

Pe = Pr Xˆ = X .

(2.129)

Theorem 2.10.1 (Fano’s Inequality) For any estimator Xˆ such that X → Y → Xˆ , with Pe = Pr(X = Xˆ ), we have

H (Pe) + Pe log |X| ≥ H (X|Xˆ ) ≥ H (X|Y ).

(2.130)

This inequality can be weakened to

1 + Pe log |X| ≥ H (X|Y )

(2.131)

or

H (X|Y ) − 1

Pe ≥

. log |X|

(2.132)

Remark Note from (2.130) that Pe = 0 implies that H (X|Y ) = 0, as intuition suggests.

Proof: We first ignore the role of Y and prove the first inequality in (2.130). We will then use the data-processing inequality to prove the more traditional form of Fano’s inequality, given by the second inequality in (2.130). Define an error random variable,

E=

1 0

if Xˆ = X, if Xˆ = X.

(2.133)

Then, using the chain rule for entropies to expand H (E, X|Xˆ ) in two different ways, we have

H (E, X|Xˆ ) = H (X|Xˆ ) + H (E|X, Xˆ )

=0
= H (E|Xˆ ) + H (X|E, Xˆ ) .

≤H (Pe)

≤Pe log |X |

(2.134) (2.135)

Since conditioning reduces entropy, H (E|Xˆ ) ≤ H (E) = H (Pe). Now since E is a function of X and Xˆ , the conditional entropy H (E|X, Xˆ ) is

2.10 FANO’S INEQUALITY 39

equal to 0. Also, since E is a binary-valued random variable, H (E) = H (Pe). The remaining term, H (X|E, Xˆ ), can be bounded as follows:

H (X|E, Xˆ ) = Pr(E = 0)H (X|Xˆ , E = 0) + Pr(E = 1)H (X|Xˆ , E = 1)

≤ (1 − Pe)0 + Pe log |X|,

(2.136)

since given E = 0, X = Xˆ , and given E = 1, we can upper bound the conditional entropy by the log of the number of possible outcomes. Combining these results, we obtain

H (Pe) + Pe log |X| ≥ H (X|Xˆ ).

(2.137)

By the data-processing inequality, we have I (X; Xˆ ) ≤ I (X; Y ) since X → Y → Xˆ is a Markov chain, and therefore H (X|Xˆ ) ≥ H (X|Y ). Thus,
we have

H (Pe) + Pe log |X| ≥ H (X|Xˆ ) ≥ H (X|Y ).

(2.138)

Corollary For any two random variables X and Y , let p = Pr(X = Y ).

H (p) + p log |X| ≥ H (X|Y ).

(2.139)

Proof: Let Xˆ = Y in Fano’s inequality.

For any two random variables X and Y , if the estimator g(Y ) takes values in the set X, we can strengthen the inequality slightly by replacing log |X| with log(|X| − 1).

Corollary Let Pe = Pr(X = Xˆ ), and let Xˆ : Y → X; then

H (Pe) + Pe log(|X| − 1) ≥ H (X|Y ).

(2.140)

Proof: The proof of the theorem goes through without change, except that

H (X|E, Xˆ ) = Pr(E = 0)H (X|Xˆ , E = 0) + Pr(E = 1)H (X|Xˆ , E = 1)

(2.141)

≤ (1 − Pe)0 + Pe log(|X| − 1),

(2.142)

since given E = 0, X = Xˆ , and given E = 1, the range of possible X outcomes is |X| − 1, we can upper bound the conditional entropy by the log(|X| − 1), the logarithm of the number of possible outcomes. Substi-
tuting this provides us with the stronger inequality.

40 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Remark Suppose that there is no knowledge of Y . Thus, X must be guessed without any information. Let X ∈ {1, 2, . . . , m} and p1 ≥ p2 ≥ · · · ≥ pm. Then the best guess of X is Xˆ = 1 and the resulting probability of error is Pe = 1 − p1. Fano’s inequality becomes

H (Pe) + Pe log(m − 1) ≥ H (X).

(2.143)

The probability mass function

(p1, p2, . . . , pm) =

1

−

Pe,

Pe m−

1

,

.

.

.

,

Pe m−

1

(2.144)

achieves this bound with equality. Thus, Fano’s inequality is sharp. While we are at it, let us introduce a new inequality relating probability
of error and entropy. Let X and X by two independent identically distributed random variables with entropy H (X). The probability at X = X is given by

Pr(X = X ) = p2(x).
x
We have the following inequality:

(2.145)

Lemma 2.10.1 If X and X are i.i.d. with entropy H (X), Pr(X = X ) ≥ 2−H (X),

(2.146)

with equality if and only if X has a uniform distribution.

Proof: Suppose that X ∼ p(x). By Jensen’s inequality, we have

2E log p(X) ≤ E2log p(X),

(2.147)

which implies that 2−H (X) = 2 p(x) log p(x) ≤

p(x)2log p(x) =

p2(x).

(2.148)

Corollary Let X, X be independent with X ∼ p(x), X ∼ r(x), x, x ∈ X. Then

Pr(X = X ) ≥ 2−H (p)−D(p||r), Pr(X = X ) ≥ 2−H (r)−D(r||p).

(2.149) (2.150)

Proof: We have

SUMMARY 41

2−H (p)−D(p||r) = 2 p(x) log p(x)+ p(x) log -pr((xx-)) = 2 p(x) log r(x) ≤ p(x)2log r(x)
= p(x)r(x) = Pr(X = X ),

(2.151) (2.152) (2.153)
(2.154) (2.155)

where the inequality follows from Jensen’s inequality and the convexity of the function f (y) = 2y.

The following telegraphic summary omits qualifying conditions.

SUMMARY
Definition The entropy H (X) of a discrete random variable X is defined by

Properties of H

H (X) = − p(x) log p(x).
x∈X

(2.156)

1. H (X) ≥ 0.
2. Hb(X) = (logb a)Ha(X). 3. (Conditioning reduces entropy) For any two random variables, X
and Y , we have

H (X|Y ) ≤ H (X)

(2.157)

with equality if and only if X and Y are independent.

4. H (X1, X2, . . . , Xn) ≤

n i=1

H (Xi),

with

equality

if

and

only

if

the

Xi are independent.

5. H (X) ≤ log | X |, with equality if and only if X is distributed uni-

formly over X.

6. H (p) is concave in p.

42 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Definition The relative entropy D(p q) of the probability mass function p with respect to the probability mass function q is defined by

D(p q) =

p(x) p(x) log .

x

q(x)

(2.158)

Definition The mutual information between two random variables X

and Y is defined as

I (X; Y ) =

p(x, y)

p(x, y) log

.

x∈X y∈Y

p(x)p(y)

(2.159)

Alternative expressions

H

(X)

=

Ep

log

1 p(X)

,

H

(X,

Y

)

=

Ep

log

1 p(X,

Y

)

,

H

(X|Y

)

=

Ep

log

1 p(X|Y

)

,

I

(X;

Y

)

=

Ep

log

p(X, Y ) p(X)p(Y

, )

D(p||q)

=

Ep

log

p(X) q(X)

.

(2.160) (2.161) (2.162) (2.163) (2.164)

Properties of D and I

1. I (X; Y ) = H (X) − H (X|Y ) = H (Y ) − H (Y |X) = H (X) + H (Y ) − H (X, Y ).
2. D(p q) ≥ 0 with equality if and only if p(x) = q(x), for all x ∈ X.
3. I (X; Y ) = D(p(x, y)||p(x)p(y)) ≥ 0, with equality if and only if p(x, y) = p(x)p(y) (i.e., X and Y are independent).
4. If | X |= m, and u is the uniform distribution over X, then D(p u) = log m − H (p).
5. D(p||q) is convex in the pair (p, q).

Chain rules

Entropy: H (X1, X2, . . . , Xn) =

n i=1

H

(Xi

|Xi

−1,

.

.

.

,

X1

).

Mutual information:

I (X1, X2, . . . , Xn; Y ) =

n i=1

I

(Xi

;

Y

|X1

,

X2,

.

.

.

,

Xi−1

).

PROBLEMS 43

Relative entropy: D(p(x, y)||q(x, y)) = D(p(x)||q(x)) + D(p(y|x)||q(y|x)).

Jensen’s inequality. If f is a convex function, then Ef (X) ≥ f (EX).

Log sum inequality. For n positive numbers, a1, a2, . . . , an and b1, b2, . . . , bn,

n i=1

ai

log

ai bi

≥

n
ai
i=1

log

n i=1

ai

n i=1

bi

(2.165)

with

equality

if

and

only

if

ai bi

=

constant.

Data-processing inequality. If X → Y → Z forms a Markov chain, I (X; Y ) ≥ I (X; Z).

Sufficient statistic. T (X) is sufficient relative to {fθ (x)} if and only if I (θ ; X) = I (θ ; T (X)) for all distributions on θ .

Fano’s inequality. Let Pe = Pr{Xˆ (Y ) = X}. Then

H (Pe) + Pe log |X| ≥ H (X|Y ).

(2.166)

Inequality. If X and X are independent and identically distributed, then

Pr(X = X ) ≥ 2−H (X),

(2.167)

PROBLEMS

2.1 Coin flips. A fair coin is flipped until the first head occurs. Let X denote the number of flips required.
(a) Find the entropy H (X) in bits. The following expressions may be useful:

∞

rn

=

1 1−r,

n=0

∞

nr n

=

r (1 − r)2 .

n=0

(b) A random variable X is drawn according to this distribution. Find an “efficient” sequence of yes–no questions of the form,

44 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

“Is X contained in the set S?” Compare H (X) to the expected number of questions required to determine X.
2.2 Entropy of functions. Let X be a random variable taking on a finite number of values. What is the (general) inequality relationship of H (X) and H (Y ) if (a) Y = 2X? (b) Y = cos X?
2.3 Minimum entropy. What is the minimum value of H (p1, . . . , pn) = H (p) as p ranges over the set of n-dimensional probability vectors? Find all p’s that achieve this minimum.
2.4 Entropy of functions of a random variable. Let X be a discrete random variable. Show that the entropy of a function of X is less than or equal to the entropy of X by justifying the following steps:

H (X, g(X)) (=a) H (X) + H (g(X) | X) (=b) H (X),
H (X, g(X)) (=c) H (g(X)) + H (X | g(X))
(d)
≥ H (g(X)).

(2.168) (2.169) (2.170) (2.171)

Thus, H (g(X)) ≤ H (X).
2.5 Zero conditional entropy. Show that if H (Y |X) = 0, then Y is a function of X [i.e., for all x with p(x) > 0, there is only one possible value of y with p(x, y) > 0].
2.6 Conditional mutual information vs. unconditional mutual information. Give examples of joint random variables X, Y , and Z such that (a) I (X; Y | Z) < I (X; Y ). (b) I (X; Y | Z) > I (X; Y ).
2.7 Coin weighing. Suppose that one has n coins, among which there may or may not be one counterfeit coin. If there is a counterfeit coin, it may be either heavier or lighter than the other coins. The coins are to be weighed by a balance. (a) Find an upper bound on the number of coins n so that k weighings will find the counterfeit coin (if any) and correctly declare it to be heavier or lighter.

PROBLEMS 45

(b) (Difficult) What is the coin- weighing strategy for k = 3 weighings and 12 coins?
2.8 Drawing with and without replacement. An urn contains r red, w white, and b black balls. Which has higher entropy, drawing k ≥ 2 balls from the urn with replacement or without replacement? Set it up and show why. (There is both a difficult way and a relatively simple way to do this.)
2.9 Metric. A function ρ(x, y) is a metric if for all x, y, • ρ(x, y) ≥ 0. • ρ(x, y) = ρ(y, x). • ρ(x, y) = 0 if and only if x = y. • ρ(x, y) + ρ(y, z) ≥ ρ(x, z). (a) Show that ρ(X, Y ) = H (X|Y ) + H (Y |X) satisfies the first, second, and fourth properties above. If we say that X = Y if there is a one-to-one function mapping from X to Y , the third property is also satisfied, and ρ(X, Y ) is a metric. (b) Verify that ρ(X, Y ) can also be expressed as

ρ(X, Y ) = H (X) + H (Y ) − 2I (X; Y ) (2.172)

= H (X, Y ) − I (X; Y )

(2.173)

= 2H (X, Y ) − H (X) − H (Y ). (2.174)

2.10 Entropy of a disjoint mixture. Let X1 and X2 be discrete random variables drawn according to probability mass functions p1(·) and p2(·) over the respective alphabets X1 = {1, 2, . . . , m} and X2 = {m + 1, . . . , n}. Let

X=

X1 with probability α, X2 with probability 1 − α.

(a) Find H (X) in terms of H (X1), H (X2), and α. (b) Maximize over α to show that 2H (X) ≤ 2H (X1) + 2H (X2) and
interpret using the notion that 2H(X) is the effective alphabet size.
2.11 Measure of correlation. Let X1 and X2 be identically distributed but not necessarily independent. Let

ρ = 1 − H (X2 | X1) . H (X1)

46 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

(a)

Show that ρ

=

I

(X1;X2) H (X1)

.

(b) Show that 0 ≤ ρ ≤ 1.

(c) When is ρ = 0?

(d) When is ρ = 1?

2.12 Example of joint entropy. Let p(x, y) be given by

y

X

0

0

I 3

I 3

0

I
3

Find: (a) H (X), H (Y ). (b) H (X | Y ), H (Y | X). (c) H (X, Y ). (d) H (Y ) − H (Y | X). (e) I (X; Y ). (f) Draw a Venn diagram for the quantities in parts (a) through (e).

2.13

Inequality .

Show

that

ln

x

≥

1−

1 x

for

x

>

0.

2.14 Entropy of a sum. Let X and Y be random variables that take
on values x1, x2, . . . , xr and y1, y2, . . . , ys, respectively. Let Z = X + Y.

(a) Show that H (Z|X) = H (Y |X). Argue that if X, Y are independent, then H (Y ) ≤ H (Z) and H (X) ≤ H (Z). Thus, the

addition of independent random variables adds uncertainty.

(b) Give an example of (necessarily dependent) random variables

in which H (X) > H (Z) and H (Y ) > H (Z).

(c) Under what conditions does H (Z) = H (X) + H (Y )?

2.15 Data processing. Let X1 → X2 → X3 → · · · → Xn form a Markov chain in this order; that is, let

p(x1, x2, . . . , xn) = p(x1)p(x2|x1) · · · p(xn|xn−1).

Reduce I (X1; X2, . . . , Xn) to its simplest form.
2.16 Bottleneck . Suppose that a (nonstationary) Markov chain starts in one of n states, necks down to k < n states, and then fans back to m > k states. Thus, X1 → X2 → X3, that is,

PROBLEMS 47

p(x1, x2, x3) = p(x1)p(x2|x1)p(x3|x2), for all x1 ∈ {1, 2, . . . , n}, x2 ∈ {1, 2, . . . , k}, x3 ∈ {1, 2, . . . , m}.
(a) Show that the dependence of X1 and X3 is limited by the bottleneck by proving that I (X1; X3) ≤ log k.
(b) Evaluate I (X1; X3) for k = 1, and conclude that no dependence can survive such a bottleneck.

2.17 Pure randomness and bent coins. Let X1, X2, . . . , Xn denote the

outcomes of independent flips of a bent coin. Thus, Pr {Xi =

1} = p, Pr {Xi = 0} = 1 − p, where p is unknown. We wish

to obtain a sequence Z1, Z2, . . . , ZK of fair coin flips from X1, X2, . . . , Xn. Toward this end, let f : Xn → {0, 1}∗ (where {0, 1}∗ = { , 0, 1, 00, 01, . . .} is the set of all finite-length binary

sequences) be a mapping f (X1, X2, . . . , Xn) = (Z1, Z2, . . . , ZK ),

where

Zi

∼

Bernoulli

(

1 2

),

and

K

may

depend

on

(X1, . . . , Xn).

In order that the sequence Z1, Z2, . . . appear to be fair coin flips,

the map f from bent coin flips to fair flips must have the prop-

erty that all 2k sequences (Z1, Z2, . . . , Zk) of a given length k

have equal probability (possibly 0), for k = 1, 2, . . .. For example,

for n = 2, the map f (01) = 0, f (10) = 1, f (00) = f (11) =

(the null string) has the property that Pr{Z1 = 1|K = 1} = Pr{Z1 =

0|K

=

1}

=

1 2

.

Give

reasons

for

the

following

inequalities:

nH (p) (=a) H (X1, . . . , Xn)
(b)
≥ H (Z1, Z2, . . . , ZK , K) (=c) H (K) + H (Z1, . . . , ZK |K) (=d) H (K) + E(K)
(e)
≥ EK.

Thus, no more than nH (p) fair coin tosses can be derived from (X1, . . . , Xn), on the average. Exhibit a good map f on sequences of length 4.
2.18 World Series. The World Series is a seven-game series that terminates as soon as either team wins four games. Let X be the random variable that represents the outcome of a World Series between teams A and B; possible values of X are AAAA, BABABAB, and BBBAAAA. Let Y be the number of games played, which ranges from 4 to 7. Assuming that A and B are equally matched and that

48 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

the games are independent, calculate H (X), H (Y ), H (Y |X), and H (X|Y ).
2.19 Infinite entropy. This problem shows that the entropy of a discrete random variable can be infinite. Let A = ∞ n=2(n log2 n)−1. [It is easy to show that A is finite by bounding the infinite sum by the integral of (x log2 x)−1.] Show that the integer-valued random variable X defined by Pr(X = n) = (An log2 n)−1 for n = 2, 3, . . ., has H (X) = +∞.
2.20 Run-length coding. Let X1, X2, . . . , Xn be (possibly dependent) binary random variables. Suppose that one calculates the run lengths R = (R1, R2, . . .) of this sequence (in order as they occur). For example, the sequence X = 0001100100 yields run lengths R = (3, 2, 2, 1, 2). Compare H (X1, X2, . . . , Xn), H (R), and H (Xn, R). Show all equalities and inequalities, and bound all the differences.
2.21 Markov’s inequality for probabilities. Let p(x) be a probability mass function. Prove, for all d ≥ 0, that

Pr {p(X) ≤ d} log 1 ≤ H (X). d

(2.175)

2.22 Logical order of ideas. Ideas have been developed in order of need and then generalized if necessary. Reorder the following ideas, strongest first, implications following: (a) Chain rule for I (X1, . . . , Xn; Y ), chain rule for D(p(x1, . . . , xn)||q(x1, x2, . . . , xn)), and chain rule for H (X1, X2, . . . , Xn). (b) D(f ||g) ≥ 0, Jensen’s inequality, I (X; Y ) ≥ 0.
2.23 Conditional mutual information. Consider a sequence of n binary random variables X1, X2, . . . , Xn. Each sequence with an even number of 1’s has probability 2−(n−1), and each sequence with an odd number of 1’s has probability 0. Find the mutual informations

I (X1; X2), I (X2; X3|X1), . . . , I (Xn−1; Xn|X1, . . . , Xn−2).

2.24 Average entropy. Let H (p) = −p log2 p − (1 − p) log2(1 − p) be the binary entropy function.

(a)

Evaluate

H

(

1 4

)

using

the

fact

that

log2 3 ≈ 1.584.

(Hint:

You

may wish to consider an experiment with four equally likely

outcomes, one of which is more interesting than the others.)

PROBLEMS 49

(b) Calculate the average entropy H (p) when the probability p is chosen uniformly in the range 0 ≤ p ≤ 1.
(c) (Optional ) Calculate the average entropy H (p1, p2, p3), where (p1, p2, p3) is a uniformly distributed probability vector. Generalize to dimension n.
2.25 Venn diagrams. There isn’t really a notion of mutual information common to three random variables. Here is one attempt at a definition: Using Venn diagrams, we can see that the mutual information common to three random variables X, Y , and Z can be defined by

I (X; Y ; Z) = I (X; Y ) − I (X; Y |Z) .

This quantity is symmetric in X, Y , and Z, despite the preceding asymmetric definition. Unfortunately, I (X; Y ; Z) is not necessarily nonnegative. Find X, Y , and Z such that I (X; Y ; Z) < 0, and prove the following two identities: (a) I (X; Y ; Z) = H (X, Y, Z) − H (X) − H (Y ) − H (Z) +
I (X; Y ) + I (Y ; Z) + I (Z; X). (b) I (X; Y ; Z) = H (X, Y, Z) − H (X, Y ) − H (Y, Z) −
H (Z, X) + H (X) + H (Y ) + H (Z).
The first identity can be understood using the Venn diagram analogy for entropy and mutual information. The second identity follows easily from the first.
2.26 Another proof of nonnegativity of relative entropy. In view of the fundamental nature of the result D(p||q) ≥ 0, we will give another proof. (a) Show that ln x ≤ x − 1 for 0 < x < ∞.
(b) Justify the following steps:

q(x)

−D(p||q) = p(x) ln

x

p(x)

≤ p(x) q(x) − 1

x

p(x)

≤ 0.

(2.176) (2.177) (2.178)

(c) What are the conditions for equality?

2.27 Grouping rule for entropy. Let p = (p1, p2, . . . , pm) be a prob-

ability distribution on m elements (i.e., pi ≥ 0 and

m i=1

pi

=

1).

50 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

Define a new distribution q on m − 1 elements as q1 = p1, q2 = p2, . . . , qm−2 = pm−2, and qm−1 = pm−1 + pm [i.e., the distribution q is the same as p on {1, 2, . . . , m − 2}, and the probability of the
last element in q is the sum of the last two probabilities of p].
Show that

H (p) = H (q) + (pm−1 + pm)H

pm−1 ,

pm

.

pm−1 + pm pm−1 + pm

(2.179)

2.28 Mixing increases entropy. Show that the entropy of the proba-

bility distribution, (p1, . . . , pi, . . . , pj , . . . , pm), is less than the

entropy

of

the

distribution

(p1,

.

.

.

,

pi +pj 2

,

.

.

.

,

pi +pj 2

,

. . . , pm). Show that in general any transfer of probability that

makes the distribution more uniform increases the entropy.

2.29 Inequalities. Let X, Y , and Z be joint random variables. Prove the following inequalities and find conditions for equality. (a) H (X, Y |Z) ≥ H (X|Z). (b) I (X, Y ; Z) ≥ I (X; Z). (c) H (X, Y, Z) − H (X, Y ) ≤ H (X, Z) − H (X). (d) I (X; Z|Y ) ≥ I (Z; Y |X) − I (Z; Y ) + I (X; Z).

2.30 Maximum entropy. Find the probability mass function p(x) that maximizes the entropy H (X) of a nonnegative integer-valued random variable X subject to the constraint

∞
EX = np(n) = A
n=0

for a fixed value A > 0. Evaluate this maximum H (X).
2.31 Conditional entropy. Under what conditions does H (X|g(Y )) = H (X|Y )?
2.32 Fano. We are given the following joint distribution on (X, Y ):

y

X

a

b

C

I

I

I

6

12

12

2

I
12

I
6

I
12

3

I
12

I
12

I
6

PROBLEMS 51

Let Xˆ (Y ) be an estimator for X (based on Y ) and let Pe = Pr{Xˆ (Y ) = X}. (a) Find the minimum probability of error estimator Xˆ (Y ) and the
associated Pe. (b) Evaluate Fano’s inequality for this problem and compare.
2.33 Fano’s inequality. Let Pr(X = i) = pi, i = 1, 2, . . . , m, and let p1 ≥ p2 ≥ p3 ≥ · · · ≥ pm. The minimal probability of error predictor of X is Xˆ = 1, with resulting probability of error Pe = 1 − p1. Maximize H (p) subject to the constraint 1 − p1 = Pe to find a bound on Pe in terms of H . This is Fano’s inequality in the absence of conditioning.
2.34 Entropy of initial conditions. Prove that H (X0|Xn) is nondecreasing with n for any Markov chain.
2.35 Relative entropy is not symmetric. Let the random variable X have three possible outcomes {a, b, c}. Consider two distributions on this random variable:

Symbol a b c

p(x)
1 2 1 4 1 4

q (x )
1 3 1 3 1 3

Calculate H (p), H (q), D(p||q), and D(q||p). Verify that in this case, D(p||q) = D(q||p).
2.36 Symmetric relative entropy. Although, as Problem 2.35 shows, D(p||q) = D(q||p) in general, there could be distributions for which equality holds. Give an example of two distributions p and q on a binary alphabet such that D(p||q) = D(q||p) (other than the trivial case p = q).
2.37 Relative entropy. Let X, Y, Z be three random variables with a joint probability mass function p(x, y, z). The relative entropy between the joint distribution and the product of the marginals is

D(p(x, y, z)||p(x)p(y)p(z)) = E log p(x, y, z) . (2.180) p(x)p(y)p(z)

Expand this in terms of entropies. When is this quantity zero?

52 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

2.38 The value of a question. Let X ∼ p(x), x = 1, 2, . . . , m. We are given a set S ⊆ {1, 2, . . . , m}. We ask whether X ∈ S and receive the answer

Y=

1 0

if X ∈ S if X ∈ S.

Suppose that Pr{X ∈ S} = α. Find the decrease in uncertainty H (X) − H (X|Y ). Apparently, any set S with a given α is as good as any other.
2.39 Entropy and pairwise independence. Let X, Y, Z be three binary Bernoulli(-12 ) random variables that are pairwise independent; that is, I (X; Y ) = I (X; Z) = I (Y ; Z) = 0.
(a) Under this constraint, what is the minimum value for H (X, Y, Z)?
(b) Give an example achieving this minimum.
2.40 Discrete entropies. Let X and Y be two independent integervalued random variables. Let X be uniformly distributed over {1, 2, . . . , 8}, and let Pr{Y = k} = 2−k, k = 1, 2, 3, . . .. (a) Find H (X). (b) Find H (Y ). (c) Find H (X + Y, X − Y ).
2.41 Random questions. One wishes to identify a random object X ∼ p(x). A question Q ∼ r(q) is asked at random according to r(q). This results in a deterministic answer A = A(x, q) ∈ {a1, a2, . . .}. Suppose that X and Q are independent. Then I (X; Q, A) is the uncertainty in X removed by the question–answer (Q, A). (a) Show that I (X; Q, A) = H (A|Q). Interpret. (b) Now suppose that two i.i.d. questions Q1, Q2, ∼ r(q) are asked, eliciting answers A1 and A2. Show that two questions are less valuable than twice a single question in the sense that I (X; Q1, A1, Q2, A2) ≤ 2I (X; Q1, A1).
2.42 Inequalities. Which of the following inequalities are generally ≥, =, ≤? Label each with ≥, =, or ≤. (a) H (5X) vs. H (X) (b) I (g(X); Y ) vs. I (X; Y ) (c) H (X0|X−1) vs. H (X0|X−1, X1) (d) H (X, Y )/(H (X) + H (Y )) vs. 1

PROBLEMS 53

2.43 Mutual information of heads and tails
(a) Consider a fair coin flip. What is the mutual information between the top and bottom sides of the coin?
(b) A six-sided fair die is rolled. What is the mutual information between the top side and the front face (the side most facing you)?

2.44 Pure randomness. We wish to use a three-sided coin to generate a fair coin toss. Let the coin X have probability mass function



 A, pA

X

=

B, C,

pB pC ,

where pA, pB, pC are unknown.

(a) How would you use two independent flips X1, X2 to generate

(if

possible)

a

Bernoulli(

1 2

)

random

variable

Z?

(b) What is the resulting maximum expected number of fair bits

generated?

2.45 Finite entropy. Show that for a discrete random variable X ∈ {1, 2, . . .}, if E log X < ∞, then H (X) < ∞.

2.46 Axiomatic definition of entropy (Difficult). If we assume certain

axioms for our measure of information, we will be forced to use a

logarithmic measure such as entropy. Shannon used this to justify

his initial definition of entropy. In this book we rely more on the

other properties of entropy rather than its axiomatic derivation to

justify its use. The following problem is considerably more difficult

than the other problems in this section.

If a sequence of symmetric functions Hm(p1, p2, . . . , pm) satisfies the following properties:

•

Normalization: H2

1 2

,

1 2

= 1,

• Continuity: H2(p, 1 − p) is a continuous function of p,

• Grouping: Hm(p1, p2, . . . , pm) = Hm−1(p1 + p2, p3, . . . , pm) + (p1 + p2)H2 -p1p-+1p-2 , -p1p-+2p-2 ,
prove that Hm must be of the form

m
Hm(p1, p2, . . . , pm) = − pi log pi,
i=1

m = 2, 3, . . . . (2.181)

54 ENTROPY, RELATIVE ENTROPY, AND MUTUAL INFORMATION

There are various other axiomatic formulations which result in the same definition of entropy. See, for example, the book by Csisza´r and Ko¨rner [149].

2.47 Entropy of a missorted file. A deck of n cards in order 1, 2, . . . , n is provided. One card is removed at random, then replaced at random. What is the entropy of the resulting deck?

2.48 Sequence length. How much information does the length of a

sequence give about the content of a sequence? Suppose that we

consider

a

Bernoulli

(

1 2

)

process

{Xi }.

Stop

the

process

when

the

first 1 appears. Let N designate this stopping time. Thus, XN is an

element of the set of all finite-length binary sequences {0, 1}∗ =

{0, 1, 00, 01, 10, 11, 000, . . . }.

(a) Find I (N ; XN ).

(b) Find H (XN |N ).

(c) Find H (XN ).

Let’s now consider a different stopping time. For this part, again

assume

that

Xi

∼

Bernoulli(

1 2

)

but

stop

at

time

N

=

6,

with

prob-

ability

1 3

and

stop

at

time

N = 12

with

probability

2 3

.

Let

this

stopping time be independent of the sequence X1X2 · · · X12.

(d) Find I (N ; XN ).

(e) Find H (XN |N ).

(f) Find H (XN ).

HISTORICAL NOTES
The concept of entropy was introduced in thermodynamics, where it was used to provide a statement of the second law of thermodynamics. Later, statistical mechanics provided a connection between thermodynamic entropy and the logarithm of the number of microstates in a macrostate of the system. This work was the crowning achievement of Boltzmann, who had the equation S = k ln W inscribed as the epitaph on his gravestone [361].
In the 1930s, Hartley introduced a logarithmic measure of information for communication. His measure was essentially the logarithm of the alphabet size. Shannon [472] was the first to define entropy and mutual information as defined in this chapter. Relative entropy was first defined by Kullback and Leibler [339]. It is known under a variety of names, including the Kullback–Leibler distance, cross entropy, information divergence, and information for discrimination, and has been studied in detail by Csisza´r [138] and Amari [22].

HISTORICAL NOTES 55
Many of the simple properties of these quantities were developed by Shannon. Fano’s inequality was proved in Fano [201]. The notion of sufficient statistic was defined by Fisher [209], and the notion of the minimal sufficient statistic was introduced by Lehmann and Scheffe´ [350]. The relationship of mutual information and sufficiency is due to Kullback [335]. The relationship between information theory and thermodynamics has been discussed extensively by Brillouin [77] and Jaynes [294].
The physics of information is a vast new subject of inquiry spawned from statistical mechanics, quantum mechanics, and information theory. The key question is how information is represented physically. Quantum channel capacity (the logarithm of the number of distinguishable preparations of a physical system) and quantum data compression [299] are well-defined problems with nice answers involving the von Neumann entropy. A new element of quantum information arises from the existence of quantum entanglement and the consequences (exhibited in Bell’s inequality) that the observed marginal distribution of physical events are not consistent with any joint distribution (no local realism). The fundamental text by Nielsen and Chuang [395] develops the theory of quantum information and the quantum counterparts to many of the results in this book. There have also been attempts to determine whether there are any fundamental physical limits to computation, including work by Bennett [47] and Bennett and Landauer [48].

CHAPTER 3
ASYMPTOTIC EQUIPARTITION PROPERTY

In information theory, the analog of the law of large numbers is the

asymptotic equipartition property (AEP). It is a direct consequence

of the weak law of large numbers. The law of large numbers states

that for independent, identically distributed (i.i.d.) random variables,

1 n

n i=1

Xi

is

close

to

its

expected

value

EX

for

large

values

of

n.

The

AEP

states

that

1 n

log

1 p(X1,X2,...,Xn)

is

close

to

the

entropy

H,

where

X1, X2, . . . , Xn are i.i.d. random variables and p(X1, X2, . . . , Xn) is the

probability of observing the sequence X1, X2, . . . , Xn. Thus, the proba-

bility p(X1, X2, . . . , Xn) assigned to an observed sequence will be close

to 2−nH .

This enables us to divide the set of all sequences into two sets, the

typical set, where the sample entropy is close to the true entropy, and the

nontypical set, which contains the other sequences. Most of our attention

will be on the typical sequences. Any property that is proved for the typical

sequences will then be true with high probability and will determine the

average behavior of a large sample.

First, an example. Let the random variable X ∈ {0, 1} have a probability

mass function defined by p(1) = p and p(0) = q. If X1, X2, . . . , Xn are

i.i.d. according to p(x), the probability of a sequence x1, x2, . . . , xn is

n i=1

p(xi ).

For

example,

the

probability

of

the

sequence

(1,

0,

1,

1,

0,

1)

is p Xi qn− Xi = p4q2. Clearly, it is not true that all 2n sequences of

length n have the same probability.

However, we might be able to predict the probability of the sequence

that we actually observe. We ask for the probability p(X1, X2, . . . , Xn) of the outcomes X1, X2, . . . , Xn, where X1, X2, . . . are i.i.d. ∼ p(x). This is
insidiously self-referential, but well defined nonetheless. Apparently, we

are asking for the probability of an event drawn according to the same

Elements of Information Theory, Second Edition, By Thomas M. Cover and Joy A. Thomas Copyright  2006 John Wiley & Sons, Inc.
57

58 ASYMPTOTIC EQUIPARTITION PROPERTY
probability distribution. Here it turns out that p(X1, X2, . . . , Xn) is close to 2−nH with high probability.
We summarize this by saying, “Almost all events are almost equally surprising.” This is a way of saying that
Pr (X1, X2, . . . , Xn) : p(X1, X2, . . . , Xn) = 2−n(H ± ) ≈ 1 (3.1)
if X1, X2, . . . , Xn are i.i.d. ∼ p(x). In the example just given, where p(X1, X2, . . . , Xn) = p Xi qn− Xi ,
we are simply saying that the number of 1’s in the sequence is close to np (with high probability), and all such sequences have (roughly) the same probability 2−nH(p). We use the idea of convergence in probability, defined as follows:
Definition (Convergence of random variables). Given a sequence of random variables, X1, X2, . . . , we say that the sequence X1, X2, . . . converges to a random variable X:
1. In probability if for every > 0, Pr{|Xn − X| > } → 0 2. In mean square if E(Xn − X)2 → 0 3. With probability 1 (also called almost surely) if Pr{limn→∞ Xn =
X} = 1

3.1 ASYMPTOTIC EQUIPARTITION PROPERTY THEOREM

The asymptotic equipartition property is formalized in the following theorem.

Theorem 3.1.1 (AEP ) If X1, X2, . . . are i.i.d. ∼ p(x), then

−

1
-
n

log

p(X1,

X2,

.

.

.

,

Xn)

→

H

(X)

in probability.

(3.2)

Proof: Functions of independent random variables are also independent random variables. Thus, since the Xi are i.i.d., so are log p(Xi). Hence, by the weak law of large numbers,

1

1

− n log p(X1, X2, . . . , Xn) = − n log p(Xi)

i

(3.3)

→ −E log p(X) in probability (3.4)

= H (X),

(3.5)

which proves the theorem.

3.1 ASYMPTOTIC EQUIPARTITION PROPERTY THEOREM 59

Definition The typical set A(n) with respect to p(x) is the set of sequences (x1, x2, . . . , xn) ∈ Xn with the property

2−n(H (X)+ ) ≤ p(x1, x2, . . . , xn) ≤ 2−n(H (X)− ).

(3.6)

As a consequence of the AEP, we can show that the set A(n) has the following properties:

Theorem 3.1.2

1. If (x1, x2, . . . , xn) ∈ A(n), then H (X) −

≤

−

1 n

log

p(x1,

x2,

.

.

.

,

xn) ≤ H (X) + .

2. Pr A(n) > 1 − for n sufficiently large.

3. A(n) ≤ 2n(H (X)+ ), where |A| denotes the number of elements in the

set A.

4. |A(n)| ≥ (1 − )2n(H (X)− ) for n sufficiently large.

Thus, the typical set has probability nearly 1, all elements of the typical
set are nearly equiprobable, and the number of elements in the typical set is nearly 2nH .

Proof: The proof of property (1) is immediate from the definition of A(n). The second property follows directly from Theorem 3.1.1, since the
probability of the event (X1, X2, . . . , Xn) ∈ A(n) tends to 1 as n → ∞.
Thus, for any δ > 0, there exists an n0 such that for all n ≥ n0, we have

Pr

−

1 n

log

p(X1,

X2,

.

.

.

,

Xn)

−

H

(X)

<

> 1 − δ.

(3.7)

Setting δ = , we obtain the second part of the theorem. The identification of δ = will conveniently simplify notation later.
To prove property (3), we write

1 = p(x)
x∈X n

≥

p(x)

x∈A(n)

≥

2−n(H (X)+ )

x∈A(n)

= 2−n(H (X)+ )|A(n)|,

(3.8) (3.9) (3.10) (3.11)

60 ASYMPTOTIC EQUIPARTITION PROPERTY

where the second inequality follows from (3.6). Hence |A(n)| ≤ 2n(H (X)+ ).

Finally, for sufficiently large n, Pr{A(n)} > 1 − , so that

1 − < Pr{A(n)}

≤

2−n(H (X)− )

x∈A(n)

= 2−n(H (X)− )|A(n)|,

where the second inequality follows from (3.6). Hence, |A(n)| ≥ (1 − )2n(H (X)− ),

which completes the proof of the properties of A(n).

(3.12)
(3.13) (3.14) (3.15)
(3.16)

3.2 CONSEQUENCES OF THE AEP: DATA COMPRESSION
Let X1, X2, . . . , Xn be independent, identically distributed random variables drawn from the probability mass function p(x). We wish to find short descriptions for such sequences of random variables. We divide all sequences in Xn into two sets: the typical set A(n) and its complement, as shown in Figure 3.1.

n:| |n elements

Non-typical set
Typical set A(n) : 2n(H + ) elements FIGURE 3.1. Typical sets and source coding.

∋ ∋

3.2 CONSEQUENCES OF THE AEP: DATA COMPRESSION 61
Non-typical set Description: n log | | + 2 bits

Typical set Description: n(H + ) + 2 bits

∋

FIGURE 3.2. Source code using the typical set.

We order all elements in each set according to some order (e.g., lexicographic order). Then we can represent each sequence of A(n) by giving the index of the sequence in the set. Since there are ≤ 2n(H+ ) sequences in A(n), the indexing requires no more than n(H + ) + 1 bits. [The extra
bit may be necessary because n(H + ) may not be an integer.] We pre-
fix all these sequences by a 0, giving a total length of ≤ n(H + ) + 2 bits to represent each sequence in A(n) (see Figure 3.2). Similarly, we can index each sequence not in A(n) by using not more than n log |X| + 1 bits. Prefixing these indices by 1, we have a code for all the sequences in Xn.
Note the following features of the above coding scheme:

• The code is one-to-one and easily decodable. The initial bit acts as a flag bit to indicate the length of the codeword that follows.
• We have used a brute-force enumeration of the atypical set A(n)c without taking into account the fact that the number of elements in A(n)c is less than the number of elements in Xn. Surprisingly, this is good enough to yield an efficient description.
• The typical sequences have short descriptions of length ≈ nH .
We use the notation xn to denote a sequence x1, x2, . . . , xn. Let l(xn) be the length of the codeword corresponding to xn. If n is sufficiently large so that Pr{A(n)} ≥ 1 − , the expected length of the codeword is

E(l(Xn)) = p(xn)l(xn)
xn

(3.17)

62 ASYMPTOTIC EQUIPARTITION PROPERTY

=

p(xn)l(xn) +

p(xn)l(xn)

xn∈A(n)

xn∈A(n)c

(3.18)

≤

p(xn)(n(H + ) + 2)

xn∈A(n)

+

p(xn)(n log |X| + 2)

xn∈A(n)c

(3.19)

= Pr A(n) (n(H + ) + 2) + Pr A(n)c (n log |X| + 2)

(3.20)

≤ n(H + ) + n(log |X|) + 2

(3.21)

= n(H + ),

(3.22)

where

=

+

log |X| +

2 n

can

be

made

arbitrarily

small

by

an

appro-

priate choice of followed by an appropriate choice of n. Hence we have

proved the following theorem.

Theorem 3.2.1 Let Xn be i.i.d. ∼ p(x). Let > 0. Then there exists a code that maps sequences xn of length n into binary strings such that the
mapping is one-to-one (and therefore invertible) and

E

1
-

l(Xn)

≤ H (X) +

n

(3.23)

for n sufficiently large.

Thus, we can represent sequences Xn using nH (X) bits on the average.

3.3 HIGH-PROBABILITY SETS AND THE TYPICAL SET

From the definition of A(n), it is clear that A(n) is a fairly small set that contains most of the probability. But from the definition, it is not clear whether it is the smallest such set. We will prove that the typical set has essentially the same number of elements as the smallest set, to first order in the exponent.

Definition For each n = 1, 2, . . . , let Bδ(n) ⊂ Xn be the smallest set with

Pr{Bδ(n)} ≥ 1 − δ.

(3.24)

3.3 HIGH-PROBABILITY SETS AND THE TYPICAL SET 63

We argue that Bδ(n) must have significant intersection with A(n) and therefore must have about as many elements. In Problem 3.3.11, we outline
the proof of the following theorem.

Theorem 3.3.1

Let

X1, X2, . . . , Xn

be

i.i.d.

∼ p(x).

For

δ

<

1 2

and

any δ > 0, if Pr{Bδ(n)} > 1 − δ, then

1 n

log |Bδ(n)|

>

H

−

δ

for n sufficiently large.

(3.25)

Thus, Bδ(n) must have at least 2nH elements, to first order in the exponent. But A(n) has 2n(H ± ) elements. Therefore, A(n) is about the same size as the smallest high-probability set.
We will now define some new notation to express equality to first order in the exponent.

Definition The notation an =. bn means

lim 1 log an = 0. n→∞ n bn

(3.26)

Thus, an =. bn implies that an and bn are equal to the first order in the exponent.
We can now restate the above results: If δn → 0 and n → 0, then

|Bδ(nn)|=. |A(nn)|=. 2nH .

(3.27)

To illustrate the difference between A(n) and Bδ(n), let us consider a Bernoulli sequence X1, X2, . . . , Xn with parameter p = 0.9. [A Bernoulli(θ ) random variable is a binary random variable that takes on
the value 1 with probability θ .] The typical sequences in this case are the
sequences in which the proportion of 1’s is close to 0.9. However, this
does not include the most likely single sequence, which is the sequence of all 1’s. The set Bδ(n) includes all the most probable sequences and therefore includes the sequence of all 1’s. Theorem 3.3.1 implies that A(n) and Bδ(n) must both contain the sequences that have about 90% 1’s, and the two sets are almost equal in size.

64 ASYMPTOTIC EQUIPARTITION PROPERTY

SUMMARY

AEP. “Almost all events are almost equally surprising.” Specifically, if X1, X2, . . . are i.i.d. ∼ p(x), then

−1 n

log

p(X1,

X2,

.

.

.

,

Xn)

→

H

(X)

in

probability.

(3.28)

Definition. The typical set A(n) is the set of sequences x1, x2, . . . , xn satisfying

2−n(H (X)+ ) ≤ p(x1, x2, . . . , xn) ≤ 2−n(H (X)− ).

(3.29)

Properties of the typical set

1. If (x1, x2, . . . , xn) ∈ A(n), then p(x1, x2, . . . , xn) = 2−n(H ± ). 2. Pr A(n) > 1 − for n sufficiently large.
3. A(n) ≤ 2n(H (X)+ ), where |A| denotes the number of elements in set A.

Definition.

an=. bn

means

that

1 n

log

an bn

→0

as

n → ∞.

Smallest probable set. Let X1, X2, . . . , Xn be i.i.d. ∼ p(x), and for

δ

<

1 2

,

let

Bδ(n)

⊂ Xn

be

the

smallest

set

such

that

Pr{Bδ(n)} ≥

1 − δ.

Then

|Bδ(n)|=. 2nH .

(3.30)

PROBLEMS

3.1 Markov’s inequality and Chebyshev’s inequality

(a) (Markov’s inequality) For any nonnegative random variable X and any t > 0, show that

Pr {X ≥ t} ≤ EX . t

(3.31)

Exhibit a random variable that achieves this inequality with equality.
(b) (Chebyshev’s inequality) Let Y be a random variable with mean µ and variance σ 2. By letting X = (Y − µ)2, show that

PROBLEMS 65

for any > 0,

Pr {|Y − µ| >

}

≤

σ2 2.

(3.32)

(c) (Weak law of large numbers) Let Z1, Z2, . . . , Zn be a sequence

of i.i.d. random variables with mean µ and variance σ 2. Let

n

Zn

=

1 n

Zi be the sample mean. Show that

i=1

Pr Zn − µ >

≤

σ2 n 2.

(3.33)

Thus, Pr Zn − µ > → 0 as n → ∞. This is known as the weak law of large numbers.
3.2 AEP and mutual information. Let (Xi, Yi) be i.i.d. ∼ p(x, y). We form the log likelihood ratio of the hypothesis that X and Y are independent vs. the hypothesis that X and Y are dependent. What is the limit of
1 p(Xn)p(Y n) n log p(Xn, Y n) ?

3.3 Piece of cake. A cake is sliced roughly in half, the largest piece being chosen each time, the other pieces discarded. We will assume that a random cut creates pieces of proportions

P=

(

2 3

,

1 3

)

with

probability

3 4

(

2 5

,

3 5

)

with

probability

1 4

Thus, for example, the first cut (and choice of largest piece) may

result

in

a

piece

of

size

3 5

.

Cutting

and

choosing

from

this

piece

might reduce it to size

3 5

2 3

at time 2, and so on. How large, to

first order in the exponent, is the piece of cake after n cuts?

3.4 AEP . Let Xi be iid ∼ p(x), x ∈ {1, 2, . . . , m}. Let µ = EX and

H =− H | ≤ }.

p(x) log p(x). Let Bn = {xn ∈

Let Xn :

An

|

1 n

= {xn ∈ Xn :

n i=1

Xi

−

µ|

|− ≤

1
n
}.

log

p(xn)

−

(a) Does Pr{Xn ∈ An} −→ 1?

(b) Does Pr{Xn ∈ An ∩ Bn} −→ 1?

66 ASYMPTOTIC EQUIPARTITION PROPERTY

(c) Show that |An ∩ Bn| ≤ 2n(H + ) for all n.

(d) Show that |An ∩ Bn| ≥

1 2

2n(H − ) for n sufficiently large.

3.5 Sets defined by probabilities. Let X1, X2, . . . be an i.i.d. sequence of discrete random variables with entropy H (X). Let

Cn(t) = {xn ∈ X n : p(xn) ≥ 2−nt }

denote the subset of n-sequences with probabilities ≥ 2−nt . (a) Show that |Cn(t)| ≤ 2nt . (b) For what values of t does P ({Xn ∈ Cn(t)}) → 1?
3.6 AEP-like limit. Let X1, X2, . . . be i.i.d. drawn according to probability mass function p(x). Find

1

lim
n→∞

(p(X1,

X2,

.

.

.

,

Xn

))

n

.

3.7 AEP and source coding. A discrete memoryless source emits a sequence of statistically independent binary digits with probabilities p(1) = 0.005 and p(0) = 0.995. The digits are taken 100 at a time and a binary codeword is provided for every sequence of 100 digits containing three or fewer 1’s.
(a) Assuming that all codewords are the same length, find the minimum length required to provide codewords for all sequences with three or fewer 1’s.
(b) Calculate the probability of observing a source sequence for which no codeword has been assigned.
(c) Use Chebyshev’s inequality to bound the probability of observing a source sequence for which no codeword has been assigned. Compare this bound with the actual probability computed in part (b).

3.8 Products. Let

  1,

X

=



2, 3,

with

probability

1 2

with

probability

1 4

with

probability

1 4

Let X1, X2, . . . be drawn i.i.d. according to this distribution. Find the limiting behavior of the product

(X1X2

·

·

·

Xn)

1 n

.

PROBLEMS 67

3.9 AEP . Let X1, X2, . . . be independent, identically distributed ran-

dom variables drawn according to the probability mass function

p(x), x ∈ {1, 2, . . . , m}. Thus, p(x1, x2, . . . , xn) =

n i=1

p(xi

).

We

know q (x1 ,

that x2, .

.−. ,n1xlno)g=p(Xni1=,1Xq2(,x.i.).,

,

Xn) → where

H q

(X) in probability. Let is another probability

mass function on {1, 2, . . . , m}.

(a)

Evaluate

lim

−

1 n

log

q (X1 ,

X2,

.

.

.

,

Xn),

where

X1, X2, . . .

are

i.i.d. ∼ p(x).

(b) Now evaluate the limit of the log likelihood ratio

1 n

log

q (X1 ,...,Xn ) p(X1,...,Xn)

when

X1, X2, . . .

are

i.i.d.

∼ p(x).

Thus,

the

odds favoring q are exponentially small when p is true.

3.10 Random box size.

An n-dimensional rectangular box with sides X1, X2, X3, . . . , Xn is

to be constructed. The volume is Vn =

n i=1

Xi .

The

edge

length

l

of a n-cube with the same volume as the random box is l = Vn1/n.

Let X1, X2, . . . be i.i.d. uniform random variables over the unit

interval

[0,

1].

Find

limn→∞

Vn1/n

and

compare

to

(EVn)

1 n

.

Clearly,

the expected edge length does not capture the idea of the volume

of the box. The geometric mean, rather than the arithmetic mean,

characterizes the behavior of products.

3.11 Proof of Theorem 3.3.1. This problem shows that the size of the

smallest “probable” set is about 2nH . Let X1, X2, . . . , Xn be i.i.d.

∼ p(x). Let Bδ(n) ⊂ Xn such that Pr(Bδ(n)) > 1 − δ. Fix

<

1 2

.

(a) Given any two sets A, B such that Pr(A) > 1 − 1 and Pr(B) > 1 − 2, show that Pr(A ∩ B) > 1 − 1 − 2. Hence, Pr(A(n) ∩

Bδ(n)) ≥ 1 − − δ.

(b) Justify the steps in the chain of inequalities

1 − − δ ≤ Pr(A(n) ∩ Bδ(n))

(3.34)

=

p(xn)

A(n)∩Bδ(n)

(3.35)

≤

2−n(H − )

A(n)∩Bδ(n)

= |A(n) ∩ Bδ(n)|2−n(H − )

≤ |Bδ(n)|2−n(H − ).

(3.36)
(3.37) (3.38)

(c) Complete the proof of the theorem.

68 ASYMPTOTIC EQUIPARTITION PROPERTY

3.12 Monotonic convergence of the empirical distribution.
Let pˆn denote the empirical probability mass function corresponding to X1, X2, . . . , Xn i.i.d. ∼ p(x), x ∈ X. Specifically,

pˆn(x)

=

1 n

n

I (Xi = x)

i=1

is the proportion of times that Xi = x in the first n samples, where I is the indicator function.
(a) Show for X binary that

ED(pˆ2n p) ≤ ED(pˆn p).

Thus, the expected relative entropy “distance” from the empir-

ical distribution to the true distribution decreases with sample

size.

(Hint:

Write

pˆ2n

=

1 2

pˆn

+

1 2

pˆn

and

use

the

convexity

of D.)

(b) Show for an arbitrary discrete X that

ED(pˆn p) ≤ ED(pˆn−1 p).

(Hint: Write pˆn as the average of n empirical mass functions with each of the n samples deleted in turn.)
3.13 Calculation of typical set . To clarify the notion of a typical set A(n) and the smallest set of high probability Bδ(n), we will calculate the set for a simple example. Consider a sequence of i.i.d. binary random variables, X1, X2, . . . , Xn, where the probability that Xi = 1 is 0.6 (and therefore the probability that Xi = 0 is 0.4).
(a) Calculate H (X).
(b) With n = 25 and = 0.1, which sequences fall in the typical set A(n)? What is the probability of the typical set? How many elements are there in the typical set? (This involves computation of a table of probabilities for sequences with k 1’s, 0 ≤ k ≤ 25, and finding those sequences that are in the typical set.)
(c) How many elements are there in the smallest set that has probability 0.9?
(d) How many elements are there in the intersection of the sets in parts (b) and (c)? What is the probability of this intersection?

HISTORICAL NOTES 69

n k
k

0

1

1

25

2

300

3

2300

4

12650

5

53130

6

177100

7

480700

8 1081575

9 2042975

10 3268760

11 4457400

12 5200300

13 5200300

14 4457400

15 3268760

16 2042975

17 1081575

18

480700

19

177100

20

53130

21

12650

22

2300

23

300

24

25

25

1

n pk(1 − p)n−k k
0.000000 0.000000 0.000000 0.000001 0.000007 0.000054 0.000227 0.001205 0.003121 0.013169 0.021222 0.077801 0.075967 0.267718 0.146507 0.575383 0.151086 0.846448 0.079986 0.970638 0.019891 0.997633 0.001937 0.999950 0.000047 0.000003

− 1 log p(xn) n
1.321928 1.298530 1.275131 1.251733 1.228334 1.204936 1.181537 1.158139 1.134740 1.111342 1.087943 1.064545 1.041146 1.017748 0.994349 0.970951 0.947552 0.924154 0.900755 0.877357 0.853958 0.830560 0.807161 0.783763 0.760364 0.736966

HISTORICAL NOTES
The asymptotic equipartition property (AEP) was first stated by Shannon in his original 1948 paper [472], where he proved the result for i.i.d. processes and stated the result for stationary ergodic processes. McMillan [384] and Breiman [74] proved the AEP for ergodic finite alphabet sources. The result is now referred to as the AEP or the Shannon–McMillan–Breiman theorem. Chung [101] extended the theorem to the case of countable alphabets and Moy [392], Perez [417], and Kieffer [312] proved the L1 convergence when {Xi} is continuous valued and ergodic. Barron [34] and Orey [402] proved almost sure convergence for real-valued ergodic processes; a simple sandwich argument (Algoet and Cover [20]) will be used in Section 16.8 to prove the general AEP.

CHAPTER 4
ENTROPY RATES OF A STOCHASTIC PROCESS
The asymptotic equipartition property in Chapter 3 establishes that nH (X) bits suffice on the average to describe n independent and identically distributed random variables. But what if the random variables are dependent? In particular, what if the random variables form a stationary process? We will show, just as in the i.i.d. case, that the entropy H (X1, X2, . . . , Xn) grows (asymptotically) linearly with n at a rate H (X), which we will call the entropy rate of the process. The interpretation of H (X) as the best achievable data compression will await the analysis in Chapter 5.
4.1 MARKOV CHAINS
A stochastic process {Xi} is an indexed sequence of random variables. In general, there can be an arbitrary dependence among the random variables. The process is characterized by the joint probability mass functions Pr{(X1, X2, . . . , Xn) = (x1, x2, . . . , xn)} = p(x1, x2, . . . , xn), (x1, x2, . . . , xn) ∈ Xn for n = 1, 2, . . . .
Definition A stochastic process is said to be stationary if the joint distribution of any subset of the sequence of random variables is invariant with respect to shifts in the time index; that is,
Pr{X1 = x1, X2 = x2, . . . , Xn = xn} = Pr{X1+l = x1, X2+l = x2, . . . , Xn+l = xn} (4.1)
for every n and every shift l and for all x1, x2, . . . , xn ∈ X.
Elements of Information Theory, Second Edition, By Thomas M. Cover and Joy A. Thomas Copyright  2006 John Wiley & Sons, Inc.
71

72 ENTROPY RATES OF A STOCHASTIC PROCESS

A simple example of a stochastic process with dependence is one in which each random variable depends only on the one preceding it and is conditionally independent of all the other preceding random variables. Such a process is said to be Markov.

Definition A discrete stochastic process X1, X2, . . . is said to be a Markov chain or a Markov process if for n = 1, 2, . . . ,

Pr(Xn+1 = xn+1|Xn = xn, Xn−1 = xn−1, . . . , X1 = x1) = Pr (Xn+1 = xn+1|Xn = xn)

(4.2)

for all x1, x2, . . . , xn, xn+1 ∈ X. In this case, the joint probability mass function of the random variables
can be written as

p(x1, x2, . . . , xn) = p(x1)p(x2|x1)p(x3|x2) · · · p(xn|xn−1). (4.3)

Definition The Markov chain is said to be time invariant if the conditional probability p(xn+1|xn) does not depend on n; that is, for n = 1, 2, . . . ,

Pr{Xn+1 = b|Xn = a} = Pr{X2 = b|X1 = a} for all a, b ∈ X. (4.4)

We will assume that the Markov chain is time invariant unless otherwise
stated. If {Xi} is a Markov chain, Xn is called the state at time n. A time-
invariant Markov chain is characterized by its initial state and a probability transition matrix P = [Pij ], i, j ∈ {1, 2, . . . , m}, where Pij = Pr{Xn+1 = j |Xn = i}.
If it is possible to go with positive probability from any state of the
Markov chain to any other state in a finite number of steps, the Markov
chain is said to be irreducible. If the largest common factor of the lengths
of different paths from a state to itself is 1, the Markov chain is said to
aperiodic. If the probability mass function of the random variable at time n is
p(xn), the probability mass function at time n + 1 is

p(xn+1) = p(xn)Pxnxn+1 .
xn

(4.5)

A distribution on the states such that the distribution at time n + 1 is the same as the distribution at time n is called a stationary distribution. The

4.1 MARKOV CHAINS 73

stationary distribution is so called because if the initial state of a Markov chain is drawn according to a stationary distribution, the Markov chain forms a stationary process.
If the finite-state Markov chain is irreducible and aperiodic, the stationary distribution is unique, and from any starting distribution, the distribution of Xn tends to the stationary distribution as n → ∞.

Example 4.1.1 Consider a two-state Markov chain with a probability transition matrix

P=

1−α α β 1−β

(4.6)

as shown in Figure 4.1. Let the stationary distribution be represented by a vector µ whose com-
ponents are the stationary probabilities of states 1 and 2, respectively. Then the stationary probability can be found by solving the equation µP = µ or, more simply, by balancing probabilities. For the stationary distribution, the net probability flow across any cut set in the state transition graph is zero. Applying this to Figure 4.1, we obtain

µ1α = µ2β.

(4.7)

Since µ1 + µ2 = 1, the stationary distribution is

µ1

=

α

β +

, β

µ2

=

α

α +

. β

(4.8)

If the Markov chain has an initial state drawn according to the stationary distribution, the resulting process will be stationary. The entropy of the

1−α

α

1−β

State 1

β

State 2

FIGURE 4.1. Two-state Markov chain.

74 ENTROPY RATES OF A STOCHASTIC PROCESS

state Xn at time n is

β

α

H (Xn) = H α + β , α + β .

(4.9)

However, this is not the rate at which entropy grows for H (X1, X2, . . . , Xn). The dependence among the Xi’s will take a steady toll.

4.2 ENTROPY RATE

If we have a sequence of n random variables, a natural question to ask is: How does the entropy of the sequence grow with n? We define the entropy rate as this rate of growth as follows.

Definition The entropy of a stochastic process {Xi} is defined by

H

(X)

=

lim
n→∞

1 n

H

(X1,

X2,

.

.

.

,

Xn)

(4.10)

when the limit exists. We now consider some simple examples of stochastic processes and
their corresponding entropy rates.

1. Typewriter.
Consider the case of a typewriter that has m equally likely output letters. The typewriter can produce mn sequences of length n, all of them equally likely. Hence H (X1, X2, . . . , Xn) = log mn and the entropy rate is H (X) = log m bits per symbol.

2. X1, X2, . . . are i.i.d. random variables. Then

H (X)

=

lim

H (X1, X2, . . . , Xn) n

=

lim

nH (X1) n

=

H (X1),

(4.11)

which is what one would expect for the entropy rate per symbol.

3. Sequence of independent but not identically distributed random vari-

ables. In this case,
n

H (X1, X2, . . . , Xn) = H (Xi),

(4.12)

i=1

but the H (Xi)’s are all not equal. We can choose a sequence of dis-

tributions

on

X1, X2, . . . such

that

the

limit

of

1 n

H (Xi) does not

exist. An example of such a sequence is a random binary sequence