Dec 10, 2015 the new argument, which piketty spelled out recently in the french newspaper le monde, is this. Spatial inequality and development, 2005 online research. Its impact has been crucial to the success of the voyager missions to deep space. On the inequalities in information theory 5 in most systems that deals with information theory, at least two entities are relevant. A proof of the fisher information inequality via a data processing argument abstract.
This can be expressed concisely as post processing cannot increase information. We will use the data processing property of mutual information to be proved shortly, see. All dpisatisfying dependence measures are thus proved to satisfy selfequitability. This post is not about sharing solutions to those problems in the book. 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. To your second point, you missed the next statement. This is must reading for information professionals who maintain some sort of professional literacy.
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. The strength of quantum correlations is bounded from above by tsirelsons bound. Question feed subscribe to rss question feed to subscribe to this rss feed, copy and paste this url into your rss reader. In a communication system, these are the transmitter sender and receiver. Champaign information theory, learning and big data. Search the worlds most comprehensive index of fulltext books. Informally it states that you cannot increase the information content of a quantum system by acting on it with. We will use the dataprocessing property of mutual information to be proved shortly, see. Farm and industry structure and socioeconomic conditions, also published by suny press, and coauthor with paul lasly, f.
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. Does evolution violate the data processing inequality. 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 theoretic inequalities stanford university. Epi is one of the deepest inequalities in information theory, and has a. Some fundamental inequalities mark reid and aditya menon research school of computer science. How big data is automating inequality the new york times. Thats the central theme of this years world press freedom day, on may 3. Information theory, mutual information, data processing inequality, chain rule. 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. Vershynina, recovery and the data processing inequality for quasientropies, ieee trans. Y, has been the method of choice for proving impossibility converse results in information theory. 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. This model provides an interesting interpretation to the difference between the two sides of inequality 11.
Keith payne penguin random house, 2017 the disparity between rich and poor has ramifications that extend far beyond mere financial means. Y, has been the method of choice for proving impossibility converse results in information theory and many other disciplines. Lecture notes on information theory statistics, yale university. Information processing also can be considered as some thermodynamically process. Description the outline of this lecture notes are 1. The data processing inequality and stochastic resonance. Communication lower bounds for statistical estimation.
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. 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. Communication lower bounds for statistical estimation problems via a distributed data processing inequality. The data processing inequality dpi is a fundamental feature of information theory. Information theory will help us identify these fundamental limits of data compression, tranmission and inference. Assessing inequality quantitative applications in the social. Tsirelsons bound from a generalised data processing. 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. What role does and should inequality play in national and international policymaking. 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.
The mutual entropy gets degraded when data is transmitted or processed. This criterion arises naturally as a weakened form of the wellknown data processing inequality dpi. All the essential topics in information theory are covered in detail, including. Elements of information theory edition 2 by thomas m. Foremost among these is mutual information, a quantity of central importance in information theory 5, 6. 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. But the data processing inequality doesnt say the inclusion of r1 cant increase is, r2, it only says is,r1 is,r2. The aim is certainly laudable, but is information access alone really. May 04, 2018 automating inequality how hightech tools profile, police, and punish the poor by virginia eubanks 260 pp.
Many theories on income inequality, but one answer lies in. Artificial intelligence blog data processing inequality. The correct constant turns out to be a particular hypercontractivity parameter of x,y, rather than their squared maximal correlation. What are the implications of global inequality for the world economy and multilateral governance. Application of information theory, lecture 2 joint. The main object of this book will be the behavior of large sets of discrete random. Q loginf 0 1 r p q1 is the cherno information of p. Two proofs of the fisher information inequality via data. 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. 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. Is there an analog of the data processing inequality for. Elements of information theory edition 2 available in hardcover, nook book. In this comprehensive overview, the authors address these key questions. 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.
If you look at the wikipedia article for the data processing inequality. While most textbooks on information theory first define the relative entropy by. Tsirelsons bound from a generalised data processing inequality. When the smooth minentropy is used as the relevant information measure, then the dpi follows immediately from the definition of the entropy. Consider a channel that produces y given xbased on the law p yjx shown. 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. Zamir 46 used data processing properties of the fisher information to prove.
We are hence required to consider a pair of random variables not just a. Strong dataprocessing inequalities for channels and. This book should be essential reading for students and researchers in probability theory, data science, and related fields. Information theory from a functional viewpoint mit. 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. Minimal su cient statistic is a function of all other su cient statistic maximally compresses information about in the sample dr. Despite the lack of hard evidence, there are plenty of theories. The premise is learn bayesian statistics using python, explains the math notation in terms of python code not the other way around. Application of information theory, lecture 2 nov 4, 2014 8 26.
From this point of view we have proved strengthened data processing inequality in quantum 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. Lobao is professor of rural sociology, sociology, and geography at the ohio state university. Shannon information theory, usually called just information theory was introduced in 1948, 22, by c.
The goal is to highlight the common concepts and establish concrete links between these fields that enable efficient information processing and learning. Intuitions on problems from elements of information theory. We derive the fii by applying the data processing inequality to a suitable linear model relating the measurements and the parameters. From the realm of advertising to the socalled empowering networks of cyberspace, technologies continue to develop in ways that exacerbate social inequality. Larry leistritz, and katherine meyer of beyond the amber waves of grain. On the other hand, the special case of the leaveoneout sets c c n.
Jul 04, 2011 the data processing inequality dpi is a fundamental feature of information theory. The second edition of this fundamental textbook maintains the book s tradition of clear, thoughtprovoking instruction. The data processing inequality is a nice, intuitive inequality about mutual information. The widely accessible presentation will make this book a classic that everyone in foundational data science will want to have on their bookshelf. While solving problems in elements of information theory by cover and thomas, i found this problem in the last chapter. It pithily exposes flaws in how information is used to assess everything from creditworthiness to. On hypercontractivity and a data processing inequality. By data processing inequality, by processing we are losing. Information theoretic proofs of entropy power inequalities.
Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. Information inequality presents a crisp and farreaching indictment of the data deprivation that corporate interests are inflicting on the social fabric. Dataprocessing, fano dataprocessing inequality su cient statistics fanos inequality dr. In this paper we provide the correct tight constant to a data processing inequality claimed by erkip and cover. The most influential and talkedabout book on society in the last decade. In information theory, the data processing inequality states. The notion of entropy, which is fundamental to the whole topic of this book, is introduced here. For the particular question you asked with the long markov chain, polyansky and wu have a recent paper that might be of interest. We have applied this results for proving a generalized htheorem in non equilibrium thermodynamics. 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. An intuitive proof of the data processing inequality. Strong dataprocessing inequalities for channels and bayesian networks yury polyanskiy and yihong wu abstract the data processing inequality, that is, iu.
An examination of social and economic restructuring in. 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. Kolmogorov complexity theory, also known as algorithmic information theory, was introduced with di. Wilde, recoverability for holevos justasgood fidelity, in 2018 ieee international symposium on information theory isit, colorado, usa 2018, pp. Reverse dataprocessing theorems and computational second laws. This sounded great to me, as im just not comfortable. Mutual information between continuous and discrete variables from numerical data. This can be expressed concisely as postprocessing cannot increase information. These are my personal notes from an information theory course taught by prof. The data processing inequality adam kelleher medium. Yao xie, ece587, information theory, duke university.
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. Qare independent of n, then large deviation theory gives d tvp n. Sep 08, 2014 introduction to information theory 19. This is based on the book of thomas cover elements of information theory. The information sets a lower bound on the entropy, but thats it. 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. We are hence required to consider a pair of random variables not just a single random variable. 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. On upper and lower bounds for the variance of a function of a random variable cacoullos, theophilos, the annals of probability, 1982. 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.
Gibbs inequality dpjjq 0, with equality holding if p q. In the years since the first edition of the book, information theory celebrated its. Four variable data processing inequality stack exchange. 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. Essentially knowing y or z, how much can i reduce my uncertainty about x is talking about information theory. 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. May 03, 2016 access to information is a fundamental freedom. 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. Certain inequalities in information theory and the cramerrao inequality kullback, s. A strengthened data processing inequality for the belavkin. Information theoretic inequalities amir dembo, thomas m. Here we further investigate properties of this new quantum divergence.513 1106 803 581 1266 152 1327 1292 576 594 408 1354 964 1132 477 994 396 864 984 302 727 1341 1082 175 781 754 912 52 609 3 633 201 373 266 1057 465 676 151 1517 372 1297 182 1457 976 318 84 790 70 825 748