We will end this chapter by discussing the existence of the entropy rate of a stationary information source. Generalized information measures and their applications. Chain rules for entropy, relative entropy, and mutual information. Csiszar and korners book is widely regarded as a classic in the field of information theory, providing deep insights and expert treatment of the key theoretical issues. This is based on the book of thomas cover elements of information theory 2ed. An introduction to entropy and its many roles in different branches of mathematics, especially information theory, probability, combinatorics and ergodic theory. Generalizations of fanos inequality for conditional.
Elements of information theory, 2nd edition information. The book has been organized in four chapters which have each of them a different character. An important part of this book deals with geometric inequalities and this fact makes a big difference with respect to most of the books that deal with this topic in the mathematical olympiad. Free information theory books download ebooks online textbooks. Itip, the software package that comes with the book, is the only software package of its kind which can prove all shannontype information inequalities. I will go office hours as soon as possible, but for now can someone please try to explain to me fanos inequality but not through math just in a logical way that makes sense. Korner, information theory, cambridge university press 2011 r.
Lecture notes on information theory by yury polyanskiy mit and yihong wu yale other useful books recommended, will not be used in an essential way. Browse other questions tagged rmationtheory randomizedalgorithms or ask your own question. Information is uncertainty and is modeled as random variables. Elements of information theory linkedin slideshare. This note will explore the basic concepts of information theory. It is highly recommended for students planning to delve into the fields of communications, data compression, and statistical signal processing.
Fanos inequality deals with the problem of knowing the value of x given that. Random variables, x, y, and z, form a markov chain denoted as x y z, if the conditional probability distribution of z depends only on y it. While numerous informationtheoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread. Some important implications of information theory in probability theory and group theory are also explained in this book. This note will cover both classical and modern topics, including information entropy, lossless data compression, binary hypothesis testing, channel coding, and lossy data compression. Jan 31, 20 dong and fan 17 extended fano s inequality and introduced the lower bounds on the mutual information between two random variables.
These are my personal notes from an information theory course taught by prof. Fanos inequality lower bounds the probability of transmission error through a. Fanos inequality and the converse to the coding theorem. Using majorization theory, fanos inequality is generalized to a broad class of information measures, which contains those of shannon and renyi. Elements of information theory edition 2 by thomas m. The use of fanos inequality says we should use the best estimator w of w. Bounds on the optimal fscore, ber, and costsensitive risk and their implications mingjie zhao. Dataprocessing, fano dataprocessing inequality su cient statistics fanos inequality dr. Principles and practice of information theory richard e. Beyond fanos inequality journal of machine learning mit.
This is based on the book of thomas cover elements of information theory 2ed 2006 in information theory there are two key concepts. They may be distributed outside this class only with the permission of the instructor. These notes have not been subjected to the usual scrutiny reserved for formal publications. I am now reading through a book to understand fanos inequality, but i remember my professor explaining it in a certain way that made it seem so logical. Dong and fan 17 extended fanos inequality and introduced the lower bounds on the mutual information between two random variables. Elements of information theory second edition thomas m. Fanos inequality is one of the most elementary, ubiquitous, and important tools in information theory. Entropy, relative entropy, and mutual information elements. This theorem says that for any random variable x, the entropy of. This theorem says that for any random variable x, the entropy of x is upper bounded by the log of the size of the alphabet.
It is used to find a lower bound on the error probability of any decoder as well as the. Relationship between entropy and mutual information. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. Yao xie, ece587, information theory, duke university. In the information theory literature, fanos inequality fano, 1961 is a well. Jun 26, 2016 for the love of physics walter lewin may 16, 2011 duration. For the love of physics walter lewin may 16, 2011 duration. We will derive an important inequality called fanos inequality. Conditional entropy and error probability princeton university. Information theory plays an indispensable role in the development of algorithmindependent impossibility results, both for. Download pdf inequalities free online new books in politics. Free information theory books download ebooks online.
An introductory guide to fanos inequality with applications in. All the essential topics in information theory are covered in detail, including. Information theory answers two fundamental questions in communication theory. Fanos inequality is a result from information theory that relates the conditional entropy of a random variable \x\ relative to the correlated variable \y\ to the probability of incorrectly estimating \x\ from \y\. A first course in information theory edition 1 by raymond w. This book is an uptodate treatment of information theory for discrete random variables, which forms the foundation of the theory at large. This book is an uptodatetreatment of information theory for discrete random variables, which. We first introduce the concept of a markov chain, then state fanos inequality, and finally prove the data processing inequality, a result with many applications in classical information theory.
Entropy and ergodic theory ucla, fall 2017 summary. Before we establish fano s inequality, we first prove a very useful theorem. Before we establish fanos inequality, we first prove a very useful theorem. Again, at the end of the lecture video, you will find assignment 2 which is due 2 weeks from today. This is a graduatelevel introduction to mathematics of information theory. We also explore the parallels between the inequalities in information theory and inequalities. In information theory, fano s inequality also known as the fano converse and the fano lemma. The latest edition of this classic is updated with new problem sets and material the second edition of this fundamental textbook maintains the books tradition of clear, thoughtprovoking instruction. It is an essential tool for all information theorists. Information processing and learning spring 2012 lecture 2. The inequality resulted from an early attempt to relate the equivocation, the information theory.
The aim is to give a quick overview of many topics, emphasizing a few basic combinatorial problems that they have in. Information theory plays an indispensable role in the development of algorithmindependent impossibility results, both for communication problems and for seemingly distinct areas such as statistics and machine learning. The inequality that became known as the fano inequality pertains to a model of communications system in which a message selected from a set of \n\ possible messages is encoded into an input signal for transmission through a noisy channel and the resulting output signal is decoded into one of the same set of possible messages. Week 2 introduction chapter 2 information measures part 2. Oct 21, 2011 the inequality that became known as the fano inequality pertains to a model of communications system in which a message selected from a set of \n\ possible messages is encoded into an input signal for transmission through a noisy channel and the resulting output signal is decoded into one of the same set of possible messages. While numerous information theoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread.
Gibbs, data processing and fanos inequalities lecturer. These are all fundamental tools in information theory. Chapter 1 is dedicated to present basic inequalities. Information is digital, it is the same as transmiting 0s1s with no reference to what they represent.
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