Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. Engineering and computer science 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. The data processing inequality and stochastic resonance core.
Autoencoders, data processing inequality, intrinsic dimensionality, information. Due to the many challenges both, experimental and computational involved in whole genome gene regulatory networks. Vershynina, recovery and the data processing inequality for quasientropies, ieee trans. Quantum data processing inequality bounds the set of bipartite states that can be generated by two far apart parties under local operations. March 7, 2016 in deep belief networks, information theory by hundalhh permalink. Question feed subscribe to rss question feed to subscribe to this rss feed, copy and paste this url into your rss reader. Sending such a telegram costs only twenty ve cents.
Contents 1 entropy 1 2 asymptotic equipartition property2 3 entropy rates of a stochastic process4 4 data compression 5 5. Our observations have direct impact on the optimal design of autoencoders, the design of alternative feedforward training methods, and even in the problem of generalization. This criterion arises naturally as a weakened form of the wellknown data processing inequality dpi. The data processing inequality is a nice, intuitive inequality about mutual information. Artificial intelligence blog data processing inequality. In practice, this means that no more information can be obtained out of a set of data then was there to begin.
All dpisatisfying dependence measures are thus proved to satisfy selfequitability. Understanding autoencoders with information theoretic concepts. The application of information theory to biochemical. A large portion of this chapter is dedicated to studying data processing inequalities for points in euclidean space.
Transcriptional network structure assessment via the data. In this note we first survey known results relating various notions of contraction for a single channel. Information theory will help us identify these fundamental limits of data compression, tranmission and inference. Information theory started with claude shannons a mathematical theory of communication. Thomas courtade fall 2016 these are my personal notes from an information theory course taught by prof. Here by dataprocessing i mean application of an arbitrary bistochastic map on both arguments of the divergence and i want this to decrease the value of the divergence. Co 7392 information theory and applications university of. An intuitive proof of the data processing inequality.
Suppose x,y, z are random variables and z is independent of x given y, then. Suppose three random variables form a markov chain x. We derive the fii by applying the data processing inequality to a suitable linear model relating the measurements and the parameters. Unlike shannons mutual information and in violation of the data processing inequality, vinformation can be created through computation. As our main technique, we prove a \textitdistributed data processing inequality, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems. Reverse dataprocessing theorems and computational second laws. A standard problem in information theory and statistical inference is to understand the degrada tion of a. Yao xie, ece587, information theory, duke university. Communication lower bounds for statistical estimation.
An introduction to information theory and applications. Outline entropy and information data compression 1 entropy and information entropy information inequality data processing inequality 2 data compression asymptotic equipartition property aep typical sets noiseless source coding theorem. The data processing inequality and stochastic resonance. This inequality gives the fundamental relationship between probability density functions and pre. Information overload is a serious challenge for a variety of information systems. Jensens inequality, data processing theorem, fanoss inequality. Mutual information between continuous and discrete variables from numerical data. Check out raymond yeungs book on information theory and network coding to convert the above problem to that of set theoretic. The rst building block was entropy, which he sought as a functional h of probability densities with two desired. 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. Even for x with pdf h x can be positive, negative, take values of. Dataprocessing, fano dataprocessing inequality su cient statistics fanos inequality dr. Shannon entropy, divergence, and mutual information basic properties of entropic quantities chain rule, pinsker inequality, data processing inequality oneshot and asymptotic compression noisy coding theorem and errorcorrection codes bregman theorem, shearer lemma and applications communication complexity of set. Here by data processing i mean application of an arbitrary bistochastic map on both arguments of the divergence and i want this to decrease the value of the divergence.
But due to the additivity of quantum mutual information under tensor. Informationtheoretic methods in highdimensional statistics. Finally, we discuss the data processing inequality, which essentially states that at every step of information processing, information cannot be gained, only lost. Data processing inequality and unsurprising implications. Lecture notes on information theory preface \there is a whole book of readymade, long and convincing, lavishly composed telegrams for all occasions. October 2012 outline contents 1 entropy and its properties 1. Entropy, relative entropy, and mutual information some basic notions of information theory radu trmbit. This inequality will seem obvious to those who know information. A great many important inequalities in information theory are actually lower bounds for the kullbackleibler divergence. You see, what gets transmitted over the telegraph is not the text of the telegram, but simply the number under which it is listed in the book. A strengthened data processing inequality for the belavkin. Mutual information, data processing inequality, chain rule. Communication lower bounds for statistical estimation problems via a distributed data processing inequality. Relationship between entropy and mutual information.
A proof of the fisher information inequality via a data. This model provides an interesting interpretation to the difference between the two sides of inequality 11. Information theory, in the technical sense, as it is used today goes back to the work. Lecture notes for statistics 311electrical engineering 377. In this work we have shown the relevance of the use of a theorem form information theory, the data processing inequality theorem 1 in the context of primary assessment of gene regulatory networks. Jan 22, 2020 shannon entropy, divergence, and mutual information basic properties of entropic quantities chain rule, pinsker inequality, data processing inequality oneshot and asymptotic compression noisy coding theorem and errorcorrection codes bregman theorem, shearer lemma and applications communication complexity of set. Generally speaking, a data processing inequality says that the amount of information between two objects cannot be significantly increased when one of the objects is processed by a particular type of transformation. Chain rules for entropy, relative entropy, and mutual information. This can be expressed concisely as post processing cannot increase information. Consider a channel that produces y given xbased on the law p yjx shown. We also explore the parallels between the inequalities in information theory and inequalities. More precisely, for a markov chain x y z, the data processing inequality states that ix. 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. The data processing inequality dpi is a fundamental feature of information theory.
Jun 07, 2009 lets suppose i have a speech signal with frequency content 300 hz. Z, then the mutual information between x and y is greater than or equal to the mutual information between x and z. Readers are provided once again with an instructive mix of mathematics, physics, statistics, and information theory. The data processing inequality guarantees that computing on data cannot increase its mutual infor.
The data processing inequality theorem of information theory states that no more information can be obtained out of a set of data then was there to begin with mcdonnell et al. Four variable data processing inequality stack exchange. Various channeldependent improvements called strong dataprocessing inequalities, or sdpis of this inequality have been proposed both classically and more recently. Today, i want to talk about how it was so successful partially from an information theoretic perspective and some lessons that we all should be aware of. A theory of usable information under computational. Foremost among these is mutual information, a quantity of central importance in information theory 5, 6. Mutual dimension, data processing inequalities, and.
Stochastic resonance and data processing inequality. Anyone know a reference that demonstrates that classical renyi divergence satisfies a data processing inequality dpi. The notion of entropy, which is fundamental to the whole topic of this book, is introduced here. Sep 25, 2019 the resulting predictive vinformation encompasses mutual information and other notions of informativeness such as the coefficient of determination.
This set of lecture notes explores some of the many connections relating information theory, statistics, computation, and learning. All the essential topics in information theory are covered in detail, including. Signal processing, machine learning, and statistics all revolve around extracting useful information from signals and data. Wilde, recoverability for holevos justasgood fidelity, in 2018 ieee international symposium on information theory isit, colorado, usa 2018, pp. Informally it states that you cannot increase the information content of a quantum system by acting on it with. I then add noise to this signal, that happens to be somewhere below 300 hz.
Dataprocessing, fano data processing inequality su cient statistics fanos inequality dr. The data processing inequality of information theory states that given random variables x, y and z that form a markov chain in the order x. 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 have heard enough about the great success of neural networks and how they are used in real problems. 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. Lets suppose i have a speech signal with frequency content 300 hz. Strong dataprocessing inequalities for channels and.