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