Intuitions on problems from elements of information theory. We establish a connection between this bound and the fact that correlations between two systems cannot increase under local operations, a property known as the data processing inequality. Suppose x,y, z are random variables and z is independent of x given y, then mix,z information theory studies the quantification, storage, and communication of information. All dpisatisfying dependence measures are thus proved to satisfy selfequitability. Tsirelsons bound from a generalised data processing inequality. The data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical. Information inequality presents a telling account of the current shift in the information landscape from a model of social accountability to a more privatized corporate model. Despite the lack of hard evidence, there are plenty of theories.
A proof of the fisher information inequality via a data processing argument abstract. The most influential and talkedabout book on society in the last decade. In this sense, zamirs data processing inequality for fisher information merely pointed out the fact that fisher information bears the real meaning as an information quantity. Farm and industry structure and socioeconomic conditions, also published by suny press, and coauthor with paul lasly, f. The mutual entropy gets degraded when data is transmitted or processed. Essentially knowing y or z, how much can i reduce my uncertainty about x is talking about information theory. This model provides an interesting interpretation to the difference between the two sides of inequality 11. Multiple approaches to assessing inequality are introduced by starting with the basic distributional property and providing connections among approaches this supplementary text is appropriate for any graduatelevel, intermediate, or advanced statistics course across the social and behavioral sciences, as well as individual researchers. Lecture notes on information theory statistics, yale university. For the particular question you asked with the long markov chain, polyansky and wu have a recent paper that might be of interest. May 03, 2016 access to information is a fundamental freedom. The information sets a lower bound on the entropy, but thats it. Inequality is a major driver of middle eastern terrorism, including the islamic state attacks on paris earlier this month and western nations have themselves largely to blame for that inequality.
The data processing inequality dpi is a fundamental feature of information theory. Application of information theory, lecture 2 nov 4, 2014 8 26. Communication lower bounds for statistical estimation problems via a distributed data processing inequality. Consider a channel that produces y given xbased on the law p yjx shown. The fisher information jx of a random variable x under a translation parameter appears in information theory in the classical proof of the entropypower inequality epi. Minimal su cient statistic is a function of all other su cient statistic maximally compresses information about in the sample dr.
Elements of information theory edition 2 by thomas m. Yao xie, ece587, information theory, duke university. Inequality has profound consequences for how we think, how we respond to stress, how our immune systems function, and even how we view justice, fairness, and other moral concepts. Information theory, mutual information, data processing inequality, chain rule. Zamir 46 used data processing properties of the fisher information to prove.
Kate pickett and richard wilkinsons the spirit level not only changed the way we understand and view inequality, it inspired the creation of the equality trust and our work. Information theoretic proofs of entropy power inequalities. The second edition of this fundamental textbook maintains the book s tradition of clear, thoughtprovoking instruction. By increased the mutual information i assume you mean, increased the mutual information between the signal and the output of the highpass filter, by adding the noise. Keith payne penguin random house, 2017 the disparity between rich and poor has ramifications that extend far beyond mere financial means. It pithily exposes flaws in how information is used to assess everything from creditworthiness to. From the realm of advertising to the socalled empowering networks of cyberspace, technologies continue to develop in ways that exacerbate social inequality. Y, has been the method of choice for proving impossibility converse results in information theory and many other disciplines. Question feed subscribe to rss question feed to subscribe to this rss feed, copy and paste this url into your rss reader.
Information processing also can be considered as some thermodynamically process. We have applied this results for proving a generalized htheorem in non equilibrium thermodynamics. All the essential topics in information theory are covered in detail, including. An intuitive proof of the data processing inequality. Many theories on income inequality, but one answer lies in. The data processing inequality is an information theoretic concept which states that the information content of a signal cannot be increased via a local physical operation. Application of information theory, lecture 2 joint. Does evolution violate the data processing inequality.
Champaign information theory, learning and big data. To your second point, you missed the next statement. The correct constant turns out to be a particular hypercontractivity parameter of x,y, rather than their squared maximal correlation. What role does and should inequality play in national and international policymaking. The goal is to highlight the common concepts and establish concrete links between these fields that enable efficient information processing and learning. As the title indicates, this post is rather about sharing intuitions or interpretations of some results mentioned or alluded in some problems listed at the end of chapter 2 of the book elements of information theory. The widely accessible presentation will make this book a classic that everyone in foundational data science will want to have on their bookshelf.
What are the implications of global inequality for the world economy and multilateral governance. These are my personal notes from an information theory course taught by prof. On hypercontractivity and a data processing inequality. May 04, 2018 automating inequality how hightech tools profile, police, and punish the poor by virginia eubanks 260 pp. But the data processing inequality doesnt say the inclusion of r1 cant increase is, r2, it only says is,r1 is,r2.
In this paper we provide the correct tight constant to a data processing inequality claimed by erkip and cover. How big data increases inequality and threatens democracy the crown publishing group is your community for cooking, recipes, and the inside scoop on your favorite cookbook authors. Strong dataprocessing inequalities for channels and. Alfred hero, university of michigan vershynins book is a brilliant introduction to the mathematics which is at the core of modern signal processing and data. If you look at the wikipedia article for the data processing inequality. Certain inequalities in information theory and the cramerrao inequality kullback, s. Jun 07, 2009 by increased the mutual information i assume you mean, increased the mutual information between the signal and the output of the highpass filter, by adding the noise. Tsirelsons bound from a generalised data processing. Mutual information between continuous and discrete variables from numerical data.
Information theoretic inequalities stanford university. Assessing inequality quantitative applications in the social. Information inequality presents a crisp and farreaching indictment of the data deprivation that corporate interests are inflicting on the social fabric. While solving problems in elements of information theory by cover and thomas, i found this problem in the last chapter.
Epi is one of the deepest inequalities in information theory, and has a. The main object of this book will be the behavior of large sets of discrete random. Its impact has been crucial to the success of the voyager missions to deep space. Reverse dataprocessing theorems and computational second laws. We are hence required to consider a pair of random variables not just a. Strong dataprocessing inequalities for channels and bayesian networks yury polyanskiy and yihong wu abstract the data processing inequality, that is, iu. Larry leistritz, and katherine meyer of beyond the amber waves of grain. How big data is automating inequality the new york times.
When the smooth minentropy is used as the relevant information measure, then the dpi follows immediately from the definition of the entropy. Ramon van handel, princeton university, new jersey this very welcome contribution to the literature gives a concise introduction to several topics in highdimensional probability that are of key relevance in. While most textbooks on information theory first define the relative entropy by. This is based on the book of thomas cover elements of information theory. From this point of view we have proved strengthened data processing inequality in quantum information theory. Thats the central theme of this years world press freedom day, on may 3. Wilde, recoverability for holevos justasgood fidelity, in 2018 ieee international symposium on information theory isit, colorado, usa 2018, pp. We will use the data processing property of mutual information to be proved shortly, see. This book should be essential reading for students and researchers in probability theory, data science, and related fields.
In a communication system, these are the transmitter sender and receiver. On the other hand, the special case of the leaveoneout sets c c n. Lobao is professor of rural sociology, sociology, and geography at the ohio state university. Search the worlds most comprehensive index of fulltext books. Jul 04, 2011 the data processing inequality dpi is a fundamental feature of information theory. Sep 21, 2006 it is widely recognized that income inequality increased in the 1990s, but nobody knows quite why. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. Spatial inequality and development, 2005 online research.
On the inequalities in information theory 5 in most systems that deals with information theory, at least two entities are relevant. Elements of information theory edition 2 available in hardcover, nook book. Check out raymond yeungs book on information theory and network coding to convert the above problem to that of set theoretic and measure theoretic manipulation. The data processing inequality and stochastic resonance. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. The aim is certainly laudable, but is information access alone really. Some fundamental inequalities mark reid and aditya menon research school of computer science. Informally it states that you cannot increase the information content of a quantum system by acting on it with. We derive the fii by applying the data processing inequality to a suitable linear model relating the measurements and the parameters.
Description the outline of this lecture notes are 1. This sounded great to me, as im just not comfortable. Artificial intelligence blog data processing inequality. Many developing and transition countries exhibit significant regional disparities in average incomes, the incidence and depth of poverty, health indicators, education status, and other correlates of living standards and human development. The notion of entropy, which is fundamental to the whole topic of this book, is introduced here. By data processing inequality, by processing we are losing. Shannon information theory, usually called just information theory was introduced in 1948, 22, by c. The strength of quantum correlations is bounded from above by tsirelsons bound. Qare independent of n, then large deviation theory gives d tvp n. Here we further investigate properties of this new quantum divergence.
This post is not about sharing solutions to those problems in the book. Data processing is a general principle in information theory, in that any quantity under the name information should obey some sort of data processing inequality. This criterion arises naturally as a weakened form of the wellknown data processing inequality dpi. Y, has been the method of choice for proving impossibility converse results in information theory. An examination of social and economic restructuring in. The premise is learn bayesian statistics using python, explains the math notation in terms of python code not the other way around. On upper and lower bounds for the variance of a function of a random variable cacoullos, theophilos, the annals of probability, 1982. Communication lower bounds for statistical estimation. The data processing inequality is a nice, intuitive inequality about mutual information. A proof of the fisher information inequality via a data. The notion of entropy, which is fundamental to the whole topic of this book.
Information theory from a functional viewpoint mit. The data processing inequality adam kelleher medium. In the years since the first edition of the book, information theory celebrated its. Information theoretic inequalities amir dembo, thomas m. It was originally proposed by claude shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled a mathematical theory of communication. This can be expressed concisely as postprocessing cannot increase information. Foremost among these is mutual information, a quantity of central importance in information theory 5, 6. This is must reading for information professionals who maintain some sort of professional literacy. Dataprocessing, fano dataprocessing inequality su cient statistics fanos inequality dr. In this comprehensive overview, the authors address these key questions.
Is there an analog of the data processing inequality for. Suppose x,y, z are random variables and z is independent of x given y, then mix,z data processing inequalities for channels and bayesian networks yury polyanskiy and yihong wu abstract the data processing inequality, that is, iu. Q loginf 0 1 r p q1 is the cherno information of p. In information theory, the data processing inequality states. We are hence required to consider a pair of random variables not just a single random variable. Even the shannontype inequalities can be considered part of this category, since the bivariate mutual information can be expressed as the kullbackleibler divergence of the joint distribution with respect to the product of the marginals, and thus these inequalities can be seen as a special case of gibbs inequality. We also explore the parallels between the inequalities in information theory and inequalities in other branches of mathematics such as matrix theory and probability theory. Dec 10, 2015 the new argument, which piketty spelled out recently in the french newspaper le monde, is this. Kolmogorov complexity theory, also known as algorithmic information theory, was introduced with di. Four variable data processing inequality stack exchange.
Vershynina, recovery and the data processing inequality for quasientropies, ieee trans. Two proofs of the fisher information inequality via data. This can be expressed concisely as post processing cannot increase information. We will use the dataprocessing property of mutual information to be proved shortly, see. Gibbs inequality dpjjq 0, with equality holding if p q. 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. Information theory will help us identify these fundamental limits of data compression, tranmission and inference. Dataprocessing, fano data processing inequality su cient statistics fanos inequality dr. Sep 08, 2014 introduction to information theory 19. This new quantum divergence has already found applications in quantum information theory. May 07, 2003 the data processing inequality of information theory states that given random variables x, y and z which form a markov chain in the order xyz, then the mutual information between x and y is greater than or equal to the mutual information between x and z.
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