A Non-Parametric Differential Entropy Rate Estimator Andrew Feutrill and Matthew Roughan, Fellow, IEEE
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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. -
An Unforeseen Equivalence Between Uncertainty and Entropy
An Unforeseen Equivalence between Uncertainty and Entropy Tim Muller1 University of Nottingham [email protected] Abstract. Beta models are trust models that use a Bayesian notion of evidence. In that paradigm, evidence comes in the form of observing whether an agent has acted well or not. For Beta models, uncertainty is the inverse of the amount of evidence. Information theory, on the other hand, offers a fundamentally different approach to the issue of lacking knowledge. The entropy of a random variable is determined by the shape of its distribution, not by the evidence used to obtain it. However, we dis- cover that a specific entropy measure (EDRB) coincides with uncertainty (in the Beta model). EDRB is the expected Kullback-Leibler divergence between two Bernouilli trials with parameters randomly selected from the distribution. EDRB allows us to apply the notion of uncertainty to other distributions that may occur when generalising Beta models. Keywords: Uncertainty · Entropy · Information Theory · Beta model · Subjective Logic 1 Introduction The Beta model paradigm is a powerful formal approach to studying trust. Bayesian logic is at the core of the Beta model: \agents with high integrity be- have honestly" becomes \honest behaviour evidences high integrity". Its simplest incarnation is to apply Beta distributions naively, and this approach has limited success. However, more powerful and sophisticated approaches are widespread (e.g. [3,13,17]). A commonality among many approaches, is that more evidence (in the form of observing instances of behaviour) yields more certainty of an opinion. Uncertainty is inversely proportional to the amount of evidence. Evidence is often used in machine learning. -
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. -
Guaranteed Bounds on Information-Theoretic Measures of Univariate Mixtures Using Piecewise Log-Sum-Exp Inequalities
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 20 October 2016 doi:10.20944/preprints201610.0086.v1 Peer-reviewed version available at Entropy 2016, 18, 442; doi:10.3390/e18120442 Article Guaranteed Bounds on Information-Theoretic Measures of Univariate Mixtures Using Piecewise Log-Sum-Exp Inequalities Frank Nielsen 1,2,* and Ke Sun 1 1 École Polytechnique, Palaiseau 91128, France; [email protected] 2 Sony Computer Science Laboratories Inc., Paris 75005, France * Correspondence: [email protected] Abstract: Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures. Keywords: information geometry; mixture models; log-sum-exp bounds 1. Introduction Mixture models are commonly used in signal processing. A typical scenario is to use mixture models [1–3] to smoothly model histograms. For example, Gaussian Mixture Models (GMMs) can be used to convert grey-valued images into binary images by building a GMM fitting the image intensity histogram and then choosing the threshold as the average of the Gaussian means [1] to binarize the image. -
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. -
Specific Differential Entropy Rate Estimation for Continuous-Valued Time Series
Specific Differential Entropy Rate Estimation for Continuous-Valued Time Series David Darmon1 1Department of Military and Emergency Medicine, Uniformed Services University, Bethesda, MD 20814, USA October 16, 2018 Abstract We introduce a method for quantifying the inherent unpredictability of a continuous-valued time series via an extension of the differential Shan- non entropy rate. Our extension, the specific entropy rate, quantifies the amount of predictive uncertainty associated with a specific state, rather than averaged over all states. We relate the specific entropy rate to pop- ular `complexity' measures such as Approximate and Sample Entropies. We provide a data-driven approach for estimating the specific entropy rate of an observed time series. Finally, we consider three case studies of estimating specific entropy rate from synthetic and physiological data relevant to the analysis of heart rate variability. 1 Introduction The analysis of time series resulting from complex systems must often be per- formed `blind': in many cases, mechanistic or phenomenological models are not available because of the inherent difficulty in formulating accurate models for complex systems. In this case, a typical analysis may assume that the data are the model, and attempt to generalize from the data in hand to the system. arXiv:1606.02615v1 [cs.LG] 8 Jun 2016 For example, a common question to ask about a times series is how `complex' it is, where we place complex in quotes to emphasize the lack of a satisfactory definition of complexity at present [60]. An answer is then sought that agrees with a particular intuition about what makes a system complex: for example, trajectories from periodic and entirely random systems appear simple, while tra- jectories from chaotic systems appear quite complicated. -
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. -
"Differential Entropy". In: Elements of Information Theory
Elements of Information Theory Thomas M. Cover, Joy A. Thomas Copyright 1991 John Wiley & Sons, Inc. Print ISBN 0-471-06259-6 Online ISBN 0-471-20061-1 Chapter 9 Differential Entropy We now introduce the concept of differential entropy, which is the entropy of a continuous random variable. Differential entropy is also related to the shortest description length, and is similar in many ways to the entropy of a discrete random variable. But there are some important differences, and there is need for some care in using the concept. 9.1 DEFINITIONS Definition: Let X be a random variable with cumulative distribution function F(x) = Pr(X I x). If F(x) is continuous, the random variable is said to be continuous. Let fix) = F’(x) when the derivative is defined. If J”co fb> = 1, th en fl x 1 is called the probability density function for X. The set where f(x) > 0 is called the support set of X. Definition: The differential entropy h(X) of a continuous random vari- able X with a density fix) is defined as h(X) = - f(x) log f(x) dx , (9.1) where S is the support set of the random variable. As in the discrete case, the differential entropy depends only on the probability density of the random variable, and hence the differential entropy is sometimes written as h(f) rather than h(X). 2.24 9.2 THE AEP FOR CONTZNUOUS RANDOM VARIABLES 225 Remark: As in every example involving an integral, or even a density, we should include the statement if it exists.