Elements of Information Theory

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 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 Cramér–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