This sounded great to me, as im just not comfortable. When the smooth minentropy is used as the relevant information measure, then the dpi follows immediately from the definition of the entropy. 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. Four variable data processing inequality stack exchange. On the other hand, the special case of the leaveoneout sets c c n. The goal is to highlight the common concepts and establish concrete links between these fields that enable efficient information processing and learning. Jul 04, 2011 the data processing inequality dpi is a fundamental feature of information theory. 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. The data processing inequality dpi is a fundamental feature of information theory. Application of information theory, lecture 2 nov 4, 2014 8 26.
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. Description the outline of this lecture notes are 1. May 04, 2018 automating inequality how hightech tools profile, police, and punish the poor by virginia eubanks 260 pp. Sep 08, 2014 introduction to information theory 19. It pithily exposes flaws in how information is used to assess everything from creditworthiness to. Many theories on income inequality, but one answer lies in. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. 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. Information processing also can be considered as some thermodynamically process. Yao xie, ece587, information theory, duke university. Information theoretic inequalities stanford university. This book should be essential reading for students and researchers in probability theory, data science, and related fields. 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.
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. Information theory will help us identify these fundamental limits of data compression, tranmission and inference. 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. Kolmogorov complexity theory, also known as algorithmic information theory, was introduced with di. 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. Zamir 46 used data processing properties of the fisher information to prove.
The data processing inequality adam kelleher medium. Information theory from a functional viewpoint mit. Larry leistritz, and katherine meyer of beyond the amber waves of grain. Dataprocessing, fano dataprocessing inequality su cient statistics fanos inequality dr. Information theoretic inequalities amir dembo, thomas m. If you look at the wikipedia article for the data processing inequality. The notion of entropy, which is fundamental to the whole topic of this book, is introduced here. All dpisatisfying dependence measures are thus proved to satisfy selfequitability.
Strong dataprocessing inequalities for channels and bayesian networks yury polyanskiy and yihong wu abstract the data processing inequality, that is, iu. Epi is one of the deepest inequalities in information theory, and has a. While solving problems in elements of information theory by cover and thomas, i found this problem in the last chapter. In the years since the first edition of the book, information theory celebrated its. Y, has been the method of choice for proving impossibility converse results in information theory. 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. From this point of view we have proved strengthened data processing inequality in quantum information theory. On hypercontractivity and a data processing inequality. A proof of the fisher information inequality via a data processing argument abstract. This is based on the book of thomas cover elements of information theory. In a communication system, these are the transmitter sender and receiver.
Sep 21, 2006 it is widely recognized that income inequality increased in the 1990s, but nobody knows quite why. Shannon information theory, usually called just information theory was introduced in 1948, 22, by c. The most influential and talkedabout book on society in the last decade. The data processing inequality is a nice, intuitive inequality about mutual information. Elements of information theory edition 2 available in hardcover, nook book. Keith payne penguin random house, 2017 the disparity between rich and poor has ramifications that extend far beyond mere financial means.
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. Application of information theory, lecture 2 joint. Reverse dataprocessing theorems and computational second laws. We derive the fii by applying the data processing inequality to a suitable linear model relating the measurements and the parameters. 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 mutual entropy gets degraded when data is transmitted or processed. An examination of social and economic restructuring in. Its impact has been crucial to the success of the voyager missions to deep space. Elements of information theory edition 2 by thomas m. The information sets a lower bound on the entropy, but thats it. On the inequalities in information theory 5 in most systems that deals with information theory, at least two entities are relevant. The correct constant turns out to be a particular hypercontractivity parameter of x,y, rather than their squared maximal correlation.
Two proofs of the fisher information inequality via data. From the realm of advertising to the socalled empowering networks of cyberspace, technologies continue to develop in ways that exacerbate social inequality. Information theoretic proofs of entropy power inequalities. 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. Certain inequalities in information theory and the cramerrao inequality kullback, s. But the data processing inequality doesnt say the inclusion of r1 cant increase is, r2, it only says is,r1 is,r2. We are hence required to consider a pair of random variables not just a single random variable. Despite the lack of hard evidence, there are plenty of theories. 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. Search the worlds most comprehensive index of fulltext books.
The premise is learn bayesian statistics using python, explains the math notation in terms of python code not the other way around. Farm and industry structure and socioeconomic conditions, also published by suny press, and coauthor with paul lasly, f. We have applied this results for proving a generalized htheorem in non equilibrium thermodynamics. Lobao is professor of rural sociology, sociology, and geography at the ohio state university. Thats the central theme of this years world press freedom day, on may 3. 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. We are hence required to consider a pair of random variables not just a. While most textbooks on information theory first define the relative entropy by. Champaign information theory, learning and big data.
For the particular question you asked with the long markov chain, polyansky and wu have a recent paper that might be of interest. Consider a channel that produces y given xbased on the law p yjx shown. To your second point, you missed the next statement. Information inequality presents a crisp and farreaching indictment of the data deprivation that corporate interests are inflicting on the social fabric. We will use the dataprocessing property of mutual information to be proved shortly, see.
This can be expressed concisely as postprocessing cannot increase information. This model provides an interesting interpretation to the difference between the two sides of inequality 11. These are my personal notes from an information theory course taught by prof. Tsirelsons bound from a generalised data processing inequality. The aim is certainly laudable, but is information access alone really. This can be expressed concisely as post processing cannot increase information. 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.
A strengthened data processing inequality for the belavkin. Information theory, mutual information, data processing inequality, chain rule. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. All the essential topics in information theory are covered in detail, including. In information theory, the data processing inequality states.
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. Foremost among these is mutual information, a quantity of central importance in information theory 5, 6. Some fundamental inequalities mark reid and aditya menon research school of computer science. Communication lower bounds for statistical estimation problems via a distributed data processing inequality. By data processing inequality, by processing we are losing. Dec 10, 2015 the new argument, which piketty spelled out recently in the french newspaper le monde, is this. Gibbs inequality dpjjq 0, with equality holding if p q. Q loginf 0 1 r p q1 is the cherno information of p. This post is not about sharing solutions to those problems in the book. Assessing inequality quantitative applications in the social. Is there an analog of the data processing inequality for.
This new quantum divergence has already found applications in quantum 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. The data processing inequality and stochastic resonance. Lecture notes on information theory statistics, yale university. 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. Communication lower bounds for statistical estimation.
In this comprehensive overview, the authors address these key questions. 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. 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. Question feed subscribe to rss question feed to subscribe to this rss feed, copy and paste this url into your rss reader. 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 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. May 03, 2016 access to information is a fundamental freedom. Y, has been the method of choice for proving impossibility converse results in information theory and many other disciplines. What are the implications of global inequality for the world economy and multilateral governance. On upper and lower bounds for the variance of a function of a random variable cacoullos, theophilos, the annals of probability, 1982. In this paper we provide the correct tight constant to a data processing inequality claimed by erkip and cover.
Dataprocessing, fano data processing inequality su cient statistics fanos inequality dr. What role does and should inequality play in national and international policymaking. Essentially knowing y or z, how much can i reduce my uncertainty about x is talking about information theory. Artificial intelligence blog data processing inequality. A proof of the fisher information inequality via a data.
Mutual information between continuous and discrete variables from numerical data. Wilde, recoverability for holevos justasgood fidelity, in 2018 ieee international symposium on information theory isit, colorado, usa 2018, pp. 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. 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. Qare independent of n, then large deviation theory gives d tvp n. This is must reading for information professionals who maintain some sort of professional literacy. Informally it states that you cannot increase the information content of a quantum system by acting on it with. Tsirelsons bound from a generalised data processing. An intuitive proof of the data processing inequality. Minimal su cient statistic is a function of all other su cient statistic maximally compresses information about in the sample dr.
The second edition of this fundamental textbook maintains the book s tradition of clear, thoughtprovoking instruction. 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. 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. Does evolution violate the data processing inequality. Intuitions on problems from elements of information theory. The notion of entropy, which is fundamental to the whole topic of this book. Spatial inequality and development, 2005 online research. The strength of quantum correlations is bounded from above by tsirelsons bound. Vershynina, recovery and the data processing inequality for quasientropies, ieee trans. 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. The main object of this book will be the behavior of large sets of discrete random. How big data is automating inequality the new york times.
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