信息论基础：第二版

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"信息论是研究信息的量化、传输和处理的数学理论，由克劳德·香农在20世纪40年代初创立。它在通信、计算机科学、编码理论、统计学、数据压缩以及许多其他领域都有广泛的应用。《Elements of Information Theory》第二版是由Thomas M. Cover和Joy A. Thomas合著的一本经典教材，详细阐述了信息论的基础概念和技术。\n\n在信息论中，几个核心的概念包括：\n\n1. 熵（Entropy）：熵是衡量信息不确定性的度量，通常用比特（bit）表示。对于一个离散随机变量，熵定义了其每个可能值出现的概率和该值的信息量的加权平均。\n\n2. 互信息（Mutual Information）：互信息描述了两个随机变量之间的关联程度，它是它们联合熵与各自边缘熵的差。\n\n3. 信源编码（Source Coding）：信源编码旨在通过减少冗余来压缩信息，例如哈夫曼编码和算术编码。\n\n4. 信道编码（Channel Coding）：信道编码是为了在噪声信道上传输信息时提高可靠性，常见的有循环冗余校验（CRC）、线性分组码和卷积码。\n\n5. 诺依曼-皮尔逊定理（Neyman-Pearson Lemma）：在假设检验中，该定理提供了最大化检测概率的最优决策规则。\n\n6. 最大似然估计（Maximum Likelihood Estimation, MLE）：在参数估计问题中，MLE是一种常用的策略，选择使得观测数据最可能出现的参数值。\n\n7. 信道容量（Channel Capacity）：由香农公式定义，信道容量是给定信道条件下可以无错误传输的最大信息速率。\n\n8. 编码定理（Coding Theorem）：香农编码定理表明，存在一种编码方法，使得在长期平均意义上，可以在任意信道上以接近信道容量的速率无误地传输信息。\n\n9. 条件熵（Conditional Entropy）：当已知某些信息时，一个随机变量的不确定性可以通过条件熵来度量。\n\n10. 联合熵（Joint Entropy）：描述两个或多个随机变量共同的不确定性。\n\n这本书深入探讨了这些概念，并提供了许多实用的算法和实际应用案例。此外，还包括了关于信息理论最新发展的章节，如网络信息论和量子信息论。\n\n信息论不仅对传统的通信系统如电话和广播有深远影响，还推动了现代技术的发展，如互联网、无线通信、数据压缩（如JPEG和MP3）、密码学、机器学习和人工智能。\n\n《Elements of Information Theory》是理解和应用信息论的宝贵资源，适合于希望深入理解信息处理和通信理论的学生、研究人员和工程师。"

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, culminat-

ing 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 com-

plexity 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 bioin-

formatics, web search, wireless communication, video compression, and

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 stu-

dents of communication theory, computer science, and statistics.

There are two points to be made about the simplicities inherent in infor-

mation 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 infor-

mation are related. Thus the answers to problems in data compression

and communication admit extensive interpretation. We all know the feel-

ing 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 New-

ton’s laws and Schr

odinger’s wave equation. Who could have foreseen the

awesome philosophical interpretations of Schr

odinger’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 investigat-

ing 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 uni-

versality 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 the-

oretic inequalities.

Even though the ﬁeld of information theory has grown considerably

since Shannon’s original paper, we have strived to emphasize its coher-

ence. While it is clear that Shannon was motivated by problems in commu-

nication theory when he developed information theory, we treat informa-

tion 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 prob-

lems. 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 inequal-

ities 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

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