On the Entropy of Sums
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Lecture 3: Entropy, Relative Entropy, and Mutual Information 1 Notation 2
EE376A/STATS376A Information Theory Lecture 3 - 01/16/2018 Lecture 3: Entropy, Relative Entropy, and Mutual Information Lecturer: Tsachy Weissman Scribe: Yicheng An, Melody Guan, Jacob Rebec, John Sholar In this lecture, we will introduce certain key measures of information, that play crucial roles in theoretical and operational characterizations throughout the course. These include the entropy, the mutual information, and the relative entropy. We will also exhibit some key properties exhibited by these information measures. 1 Notation A quick summary of the notation 1. Discrete Random Variable: U 2. Alphabet: U = fu1; u2; :::; uM g (An alphabet of size M) 3. Specific Value: u; u1; etc. For discrete random variables, we will write (interchangeably) P (U = u), PU (u) or most often just, p(u) Similarly, for a pair of random variables X; Y we write P (X = x j Y = y), PXjY (x j y) or p(x j y) 2 Entropy Definition 1. \Surprise" Function: 1 S(u) log (1) , p(u) A lower probability of u translates to a greater \surprise" that it occurs. Note here that we use log to mean log2 by default, rather than the natural log ln, as is typical in some other contexts. This is true throughout these notes: log is assumed to be log2 unless otherwise indicated. Definition 2. Entropy: Let U a discrete random variable taking values in alphabet U. The entropy of U is given by: 1 X H(U) [S(U)] = log = − log (p(U)) = − p(u) log p(u) (2) , E E p(U) E u Where U represents all u values possible to the variable. -
An Introduction to Information Theory
An Introduction to Information Theory Vahid Meghdadi reference : Elements of Information Theory by Cover and Thomas September 2007 Contents 1 Entropy 2 2 Joint and conditional entropy 4 3 Mutual information 5 4 Data Compression or Source Coding 6 5 Channel capacity 8 5.1 examples . 9 5.1.1 Noiseless binary channel . 9 5.1.2 Binary symmetric channel . 9 5.1.3 Binary erasure channel . 10 5.1.4 Two fold channel . 11 6 Differential entropy 12 6.1 Relation between differential and discrete entropy . 13 6.2 joint and conditional entropy . 13 6.3 Some properties . 14 7 The Gaussian channel 15 7.1 Capacity of Gaussian channel . 15 7.2 Band limited channel . 16 7.3 Parallel Gaussian channel . 18 8 Capacity of SIMO channel 19 9 Exercise (to be completed) 22 1 1 Entropy Entropy is a measure of uncertainty of a random variable. The uncertainty or the amount of information containing in a message (or in a particular realization of a random variable) is defined as the inverse of the logarithm of its probabil- ity: log(1=PX (x)). So, less likely outcome carries more information. Let X be a discrete random variable with alphabet X and probability mass function PX (x) = PrfX = xg, x 2 X . For convenience PX (x) will be denoted by p(x). The entropy of X is defined as follows: Definition 1. The entropy H(X) of a discrete random variable is defined by 1 H(X) = E log p(x) X 1 = p(x) log (1) p(x) x2X Entropy indicates the average information contained in X. -
On Relations Between the Relative Entropy and $\Chi^ 2$-Divergence
On Relations Between the Relative Entropy and χ2-Divergence, Generalizations and Applications Tomohiro Nishiyama Igal Sason Abstract The relative entropy and chi-squared divergence are fundamental divergence measures in information theory and statistics. This paper is focused on a study of integral relations between the two divergences, the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of f-divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong data-processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains. Keywords: Relative entropy, chi-squared divergence, f-divergences, method of types, large deviations, strong data-processing inequalities, information contraction, maximal correlation, Markov chains. I. INTRODUCTION The relative entropy (also known as the Kullback–Leibler divergence [28]) and the chi-squared divergence [46] are divergence measures which play a key role in information theory, statistics, learning, signal processing and other theoretical and applied branches of mathematics. These divergence measures are fundamental in problems pertaining to source and channel coding, combinatorics and large deviations theory, goodness-of-fit and independence tests in statistics, expectation-maximization iterative algorithms for estimating a distribution from an incomplete -
Chapter 2: Entropy and Mutual Information
Chapter 2: Entropy and Mutual Information University of Illinois at Chicago ECE 534, Natasha Devroye Chapter 2 outline • Definitions • Entropy • Joint entropy, conditional entropy • Relative entropy, mutual information • Chain rules • Jensen’s inequality • Log-sum inequality • Data processing inequality • Fano’s inequality University of Illinois at Chicago ECE 534, Natasha Devroye Definitions A discrete random variable X takes on values x from the discrete alphabet . X The probability mass function (pmf) is described by p (x)=p(x) = Pr X = x , for x . X { } ∈ X University of Illinois at Chicago ECE 534, Natasha Devroye Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. 2 Probability, Entropy, and Inference Definitions Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 This chapter, and its sibling, Chapter 8, devote some time to notation. Just You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/as the White Knight fordistinguished links. between the song, the name of the song, and what the name of the song was called (Carroll, 1998), we will sometimes 2.1: Probabilities and ensembles need to be careful to distinguish between a random variable, the v23alue of the i ai pi random variable, and the proposition that asserts that the random variable x has a particular value. In any particular chapter, however, I will use the most 1 a 0.0575 a Figure 2.2. -
A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications
entropy Article A Lower Bound on the Differential Entropy of Log-Concave Random Vectors with Applications Arnaud Marsiglietti 1,* and Victoria Kostina 2 1 Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125, USA 2 Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA; [email protected] * Correspondence: [email protected] Received: 18 January 2018; Accepted: 6 March 2018; Published: 9 March 2018 Abstract: We derive a lower bound on the differential entropy of a log-concave random variable X in terms of the p-th absolute moment of X. The new bound leads to a reverse entropy power inequality with an explicit constant, and to new bounds on the rate-distortion function and the channel capacity. Specifically, we study the rate-distortion function for log-concave sources and distortion measure ( ) = j − jr ≥ d x, xˆ x xˆ , with r 1, and we establish thatp the difference between the rate-distortion function and the Shannon lower bound is at most log( pe) ≈ 1.5 bits, independently of r and the target q pe distortion d. For mean-square error distortion, the difference is at most log( 2 ) ≈ 1 bit, regardless of d. We also provide bounds on the capacity of memoryless additive noise channels when the noise is log-concave. We show that the difference between the capacity of such channels and the capacity of q pe the Gaussian channel with the same noise power is at most log( 2 ) ≈ 1 bit. Our results generalize to the case of a random vector X with possibly dependent coordinates. -
Noisy Channel Coding
Noisy Channel Coding: Correlated Random Variables & Communication over a Noisy Channel Toni Hirvonen Helsinki University of Technology Laboratory of Acoustics and Audio Signal Processing [email protected] T-61.182 Special Course in Information Science II / Spring 2004 1 Contents • More entropy definitions { joint & conditional entropy { mutual information • Communication over a noisy channel { overview { information conveyed by a channel { noisy channel coding theorem 2 Joint Entropy Joint entropy of X; Y is: 1 H(X; Y ) = P (x; y) log P (x; y) xy X Y 2AXA Entropy is additive for independent random variables: H(X; Y ) = H(X) + H(Y ) iff P (x; y) = P (x)P (y) 3 Conditional Entropy Conditional entropy of X given Y is: 1 1 H(XjY ) = P (y) P (xjy) log = P (x; y) log P (xjy) P (xjy) y2A "x2A # y2A A XY XX XX Y It measures the average uncertainty (i.e. information content) that remains about x when y is known. 4 Mutual Information Mutual information between X and Y is: I(Y ; X) = I(X; Y ) = H(X) − H(XjY ) ≥ 0 It measures the average reduction in uncertainty about x that results from learning the value of y, or vice versa. Conditional mutual information between X and Y given Z is: I(Y ; XjZ) = H(XjZ) − H(XjY; Z) 5 Breakdown of Entropy Entropy relations: Chain rule of entropy: H(X; Y ) = H(X) + H(Y jX) = H(Y ) + H(XjY ) 6 Noisy Channel: Overview • Real-life communication channels are hopelessly noisy i.e. introduce transmission errors • However, a solution can be achieved { the aim of source coding is to remove redundancy from the source data -
Networked Distributed Source Coding
Networked distributed source coding Shizheng Li and Aditya Ramamoorthy Abstract The data sensed by differentsensors in a sensor network is typically corre- lated. A natural question is whether the data correlation can be exploited in innova- tive ways along with network information transfer techniques to design efficient and distributed schemes for the operation of such networks. This necessarily involves a coupling between the issues of compression and networked data transmission, that have usually been considered separately. In this work we review the basics of classi- cal distributed source coding and discuss some practical code design techniques for it. We argue that the network introduces several new dimensions to the problem of distributed source coding. The compression rates and the network information flow constrain each other in intricate ways. In particular, we show that network coding is often required for optimally combining distributed source coding and network information transfer and discuss the associated issues in detail. We also examine the problem of resource allocation in the context of distributed source coding over networks. 1 Introduction There are various instances of problems where correlated sources need to be trans- mitted over networks, e.g., a large scale sensor network deployed for temperature or humidity monitoring over a large field or for habitat monitoring in a jungle. This is an example of a network information transfer problem with correlated sources. A natural question is whether the data correlation can be exploited in innovative ways along with network information transfer techniques to design efficient and distributed schemes for the operation of such networks. This necessarily involves a Shizheng Li · Aditya Ramamoorthy Iowa State University, Ames, IA, U.S.A. -
On Measures of Entropy and Information
On Measures of Entropy and Information Tech. Note 009 v0.7 http://threeplusone.com/info Gavin E. Crooks 2018-09-22 Contents 5 Csiszar´ f-divergences 12 Csiszar´ f-divergence ................ 12 0 Notes on notation and nomenclature 2 Dual f-divergence .................. 12 Symmetric f-divergences .............. 12 1 Entropy 3 K-divergence ..................... 12 Entropy ........................ 3 Fidelity ........................ 12 Joint entropy ..................... 3 Marginal entropy .................. 3 Hellinger discrimination .............. 12 Conditional entropy ................. 3 Pearson divergence ................. 14 Neyman divergence ................. 14 2 Mutual information 3 LeCam discrimination ............... 14 Mutual information ................. 3 Skewed K-divergence ................ 14 Multivariate mutual information ......... 4 Alpha-Jensen-Shannon-entropy .......... 14 Interaction information ............... 5 Conditional mutual information ......... 5 6 Chernoff divergence 14 Binding information ................ 6 Chernoff divergence ................. 14 Residual entropy .................. 6 Chernoff coefficient ................. 14 Total correlation ................... 6 Renyi´ divergence .................. 15 Lautum information ................ 6 Alpha-divergence .................. 15 Uncertainty coefficient ............... 7 Cressie-Read divergence .............. 15 Tsallis divergence .................. 15 3 Relative entropy 7 Sharma-Mittal divergence ............. 15 Relative entropy ................... 7 Cross entropy -
Information Theory and Maximum Entropy 8.1 Fundamentals of Information Theory
NEU 560: Statistical Modeling and Analysis of Neural Data Spring 2018 Lecture 8: Information Theory and Maximum Entropy Lecturer: Mike Morais Scribes: 8.1 Fundamentals of Information theory Information theory started with Claude Shannon's A mathematical theory of communication. The first building block was entropy, which he sought as a functional H(·) of probability densities with two desired properties: 1. Decreasing in P (X), such that if P (X1) < P (X2), then h(P (X1)) > h(P (X2)). 2. Independent variables add, such that if X and Y are independent, then H(P (X; Y )) = H(P (X)) + H(P (Y )). These are only satisfied for − log(·). Think of it as a \surprise" function. Definition 8.1 (Entropy) The entropy of a random variable is the amount of information needed to fully describe it; alternate interpretations: average number of yes/no questions needed to identify X, how uncertain you are about X? X H(X) = − P (X) log P (X) = −EX [log P (X)] (8.1) X Average information, surprise, or uncertainty are all somewhat parsimonious plain English analogies for entropy. There are a few ways to measure entropy for multiple variables; we'll use two, X and Y . Definition 8.2 (Conditional entropy) The conditional entropy of a random variable is the entropy of one random variable conditioned on knowledge of another random variable, on average. Alternative interpretations: the average number of yes/no questions needed to identify X given knowledge of Y , on average; or How uncertain you are about X if you know Y , on average? X X h X i H(X j Y ) = P (Y )[H(P (X j Y ))] = P (Y ) − P (X j Y ) log P (X j Y ) Y Y X X = = − P (X; Y ) log P (X j Y ) X;Y = −EX;Y [log P (X j Y )] (8.2) Definition 8.3 (Joint entropy) X H(X; Y ) = − P (X; Y ) log P (X; Y ) = −EX;Y [log P (X; Y )] (8.3) X;Y 8-1 8-2 Lecture 8: Information Theory and Maximum Entropy • Bayes' rule for entropy H(X1 j X2) = H(X2 j X1) + H(X1) − H(X2) (8.4) • Chain rule of entropies n X H(Xn;Xn−1; :::X1) = H(Xn j Xn−1; :::X1) (8.5) i=1 It can be useful to think about these interrelated concepts with a so-called information diagram. -
Entropy Power Inequalities the American Institute of Mathematics
Entropy power inequalities The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Entropy power inequalities.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Fri Apr 28 17:41:04 2017 1 2 Table of Contents A.ParticipantContributions . 3 1. Anantharam, Venkatachalam 2. Bobkov, Sergey 3. Bustin, Ronit 4. Han, Guangyue 5. Jog, Varun 6. Johnson, Oliver 7. Kagan, Abram 8. Livshyts, Galyna 9. Madiman, Mokshay 10. Nayar, Piotr 11. Tkocz, Tomasz 3 Chapter A: Participant Contributions A.1 Anantharam, Venkatachalam EPIs for discrete random variables. Analogs of the EPI for intrinsic volumes. Evolution of the differential entropy under the heat equation. A.2 Bobkov, Sergey Together with Arnaud Marsliglietti we have been interested in extensions of the EPI to the more general R´enyi entropies. For a random vector X in Rn with density f, R´enyi’s entropy of order α> 0 is defined as − 2 n(α−1) α Nα(X)= f(x) dx , Z 2 which in the limit as α 1 becomes the entropy power N(x) = exp n f(x)log f(x) dx . However, the extension→ of the usual EPI cannot be of the form − R Nα(X + Y ) Nα(X)+ Nα(Y ), ≥ since the latter turns out to be false in general (even when X is normal and Y is nearly normal). Therefore, some modifications have to be done. As a natural variant, we consider proper powers of Nα. -
Information Inequalities for Joint Distributions, with Interpretations and Applications Mokshay Madiman, Member, IEEE, and Prasad Tetali, Member, IEEE
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY, 2007 1 Information Inequalities for Joint Distributions, with Interpretations and Applications Mokshay Madiman, Member, IEEE, and Prasad Tetali, Member, IEEE Abstract— Upper and lower bounds are obtained for the joint we use this flexibility later, but it is convenient to fix a entropy of a collection of random variables in terms of an default order.) arbitrary collection of subset joint entropies. These inequalities • For any set s ⊂ [n], X stands for the collection of generalize Shannon’s chain rule for entropy as well as inequalities s of Han, Fujishige and Shearer. A duality between the upper random variables (Xi : i 2 s), with the indices taken and lower bounds for joint entropy is developed. All of these in their increasing order. results are shown to be special cases of general, new results for • For any index i in [n], define the degree of i in C as submodular functions– thus, the inequalities presented constitute r(i) = jft 2 C : i 2 tgj. Let r−(s) = mini2s r(i) denote a richly structured class of Shannon-type inequalities. The new the minimal degree in s, and r (s) = max r(i) denote inequalities are applied to obtain new results in combinatorics, + i2s such as bounds on the number of independent sets in an arbitrary the maximal degree in s. graph and the number of zero-error source-channel codes, as First we present a weak form of our main inequality. well as new determinantal inequalities in matrix theory. A new inequality for relative entropies is also developed, along with Proposition I:[WEAK DEGREE FORM] Let X1;:::;Xn be interpretations in terms of hypothesis testing. -
Relative Entropy and Score Function: New Information–Estimation Relationships Through Arbitrary Additive Perturbation
Relative Entropy and Score Function: New Information–Estimation Relationships through Arbitrary Additive Perturbation Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University Evanston, IL 60208, USA Abstract—This paper establishes new information–estimation error (MMSE) of a Gaussian model [2]: relationships pertaining to models with additive noise of arbitrary d p 1 distribution. In particular, we study the change in the relative I (X; γ X + N) = mmse PX ; γ (2) entropy between two probability measures when both of them dγ 2 are perturbed by a small amount of the same additive noise. where X ∼ P and mmseP ; γ denotes the MMSE of It is shown that the rate of the change with respect to the X p X energy of the perturbation can be expressed in terms of the mean estimating X given γ X + N. The parameter γ ≥ 0 is squared difference of the score functions of the two distributions, understood as the signal-to-noise ratio (SNR) of the Gaussian and, rather surprisingly, is unrelated to the distribution of the model. By-products of formula (2) include the representation perturbation otherwise. The result holds true for the classical of the entropy, differential entropy and the non-Gaussianness relative entropy (or Kullback–Leibler distance), as well as two of its generalizations: Renyi’s´ relative entropy and the f-divergence. (measured in relative entropy) in terms of the MMSE [2]–[4]. The result generalizes a recent relationship between the relative Several generalizations and extensions of the previous results entropy and mean squared errors pertaining to Gaussian noise are found in [5]–[7].