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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

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

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

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

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 Cramer-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

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 Ω 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

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

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

17.10 Inequalities for Ratios of Determinants 683

Summary 686

Problems 686

Historical Notes 687

Bibliography 689

List of Symbols 723

Index 727