Lecture 6; Using Entropy for Evaluating and Comparing Probability Distributions Readings: Jurafsky and Martin, Section 6.7 Manning and Schutze, Section 2.2
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Entropy Rate of Stochastic Processes
Entropy Rate of Stochastic Processes Timo Mulder Jorn Peters [email protected] [email protected] February 8, 2015 The entropy rate of independent and identically distributed events can on average be encoded by H(X) bits per source symbol. However, in reality, series of events (or processes) are often randomly distributed and there can be arbitrary dependence between each event. Such processes with arbitrary dependence between variables are called stochastic processes. This report shows how to calculate the entropy rate for such processes, accompanied with some denitions, proofs and brief examples. 1 Introduction One may be interested in the uncertainty of an event or a series of events. Shannon entropy is a method to compute the uncertainty in an event. This can be used for single events or for several independent and identically distributed events. However, often events or series of events are not independent and identically distributed. In that case the events together form a stochastic process i.e. a process in which multiple outcomes are possible. These processes are omnipresent in all sorts of elds of study. As it turns out, Shannon entropy cannot directly be used to compute the uncertainty in a stochastic process. However, it is easy to extend such that it can be used. Also when a stochastic process satises certain properties, for example if the stochastic process is a Markov process, it is straightforward to compute the entropy of the process. The entropy rate of a stochastic process can be used in various ways. An example of this is given in Section 4.1. -
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. -
Entropy Rate Estimation for Markov Chains with Large State Space
Entropy Rate Estimation for Markov Chains with Large State Space Yanjun Han Department of Electrical Engineering Stanford University Stanford, CA 94305 [email protected] Jiantao Jiao Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720 [email protected] Chuan-Zheng Lee, Tsachy Weissman Yihong Wu Department of Electrical Engineering Department of Statistics and Data Science Stanford University Yale University Stanford, CA 94305 New Haven, CT 06511 {czlee, tsachy}@stanford.edu [email protected] Tiancheng Yu Department of Electronic Engineering Tsinghua University Haidian, Beijing 100084 [email protected] Abstract Entropy estimation is one of the prototypicalproblems in distribution property test- ing. To consistently estimate the Shannon entropy of a distribution on S elements with independent samples, the optimal sample complexity scales sublinearly with arXiv:1802.07889v3 [cs.LG] 24 Sep 2018 S S as Θ( log S ) as shown by Valiant and Valiant [41]. Extendingthe theory and algo- rithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with S states from a sample path of n observations. We show that Provided the Markov chain mixes not too slowly, i.e., the relaxation time is • S S2 at most O( 3 ), consistent estimation is achievable when n . ln S ≫ log S Provided the Markov chain has some slight dependency, i.e., the relaxation • 2 time is at least 1+Ω( ln S ), consistent estimation is impossible when n . √S S2 log S . Under both assumptions, the optimal estimation accuracy is shown to be S2 2 Θ( n log S ). -
Package 'Infotheo'
Package ‘infotheo’ February 20, 2015 Title Information-Theoretic Measures Version 1.2.0 Date 2014-07 Publication 2009-08-14 Author Patrick E. Meyer Description This package implements various measures of information theory based on several en- tropy estimators. Maintainer Patrick E. Meyer <[email protected]> License GPL (>= 3) URL http://homepage.meyerp.com/software Repository CRAN NeedsCompilation yes Date/Publication 2014-07-26 08:08:09 R topics documented: condentropy . .2 condinformation . .3 discretize . .4 entropy . .5 infotheo . .6 interinformation . .7 multiinformation . .8 mutinformation . .9 natstobits . 10 Index 12 1 2 condentropy condentropy conditional entropy computation Description condentropy takes two random vectors, X and Y, as input and returns the conditional entropy, H(X|Y), in nats (base e), according to the entropy estimator method. If Y is not supplied the function returns the entropy of X - see entropy. Usage condentropy(X, Y=NULL, method="emp") Arguments X data.frame denoting a random variable or random vector where columns contain variables/features and rows contain outcomes/samples. Y data.frame denoting a conditioning random variable or random vector where columns contain variables/features and rows contain outcomes/samples. method The name of the entropy estimator. The package implements four estimators : "emp", "mm", "shrink", "sg" (default:"emp") - see details. These estimators require discrete data values - see discretize. Details • "emp" : This estimator computes the entropy of the empirical probability distribution. • "mm" : This is the Miller-Madow asymptotic bias corrected empirical estimator. • "shrink" : This is a shrinkage estimate of the entropy of a Dirichlet probability distribution. • "sg" : This is the Schurmann-Grassberger estimate of the entropy of a Dirichlet probability distribution. -
Arxiv:1907.00325V5 [Cs.LG] 25 Aug 2020
Random Forests for Adaptive Nearest Neighbor Estimation of Information-Theoretic Quantities Ronan Perry1, Ronak Mehta1, Richard Guo1, Jesús Arroyo1, Mike Powell1, Hayden Helm1, Cencheng Shen1, and Joshua T. Vogelstein1;2∗ Abstract. Information-theoretic quantities, such as conditional entropy and mutual information, are critical data summaries for quantifying uncertainty. Current widely used approaches for computing such quantities rely on nearest neighbor methods and exhibit both strong performance and theoretical guarantees in certain simple scenarios. However, existing approaches fail in high-dimensional settings and when different features are measured on different scales. We propose decision forest-based adaptive nearest neighbor estimators and show that they are able to effectively estimate posterior probabilities, conditional entropies, and mutual information even in the aforementioned settings. We provide an extensive study of efficacy for classification and posterior probability estimation, and prove cer- tain forest-based approaches to be consistent estimators of the true posteriors and derived information-theoretic quantities under certain assumptions. In a real-world connectome application, we quantify the uncertainty about neuron type given various cellular features in the Drosophila larva mushroom body, a key challenge for modern neuroscience. 1 Introduction Uncertainty quantification is a fundamental desiderata of statistical inference and data science. In supervised learning settings it is common to quantify uncertainty with either conditional en- tropy or mutual information (MI). Suppose we are given a pair of random variables (X; Y ), where X is d-dimensional vector-valued and Y is a categorical variable of interest. Conditional entropy H(Y jX) measures the uncertainty in Y on average given X. On the other hand, mutual information quantifies the shared information between X and Y . -
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. -
Entropy Power, Autoregressive Models, and Mutual Information
entropy Article Entropy Power, Autoregressive Models, and Mutual Information Jerry Gibson Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106, USA; [email protected] Received: 7 August 2018; Accepted: 17 September 2018; Published: 30 September 2018 Abstract: Autoregressive processes play a major role in speech processing (linear prediction), seismic signal processing, biological signal processing, and many other applications. We consider the quantity defined by Shannon in 1948, the entropy rate power, and show that the log ratio of entropy powers equals the difference in the differential entropy of the two processes. Furthermore, we use the log ratio of entropy powers to analyze the change in mutual information as the model order is increased for autoregressive processes. We examine when we can substitute the minimum mean squared prediction error for the entropy power in the log ratio of entropy powers, thus greatly simplifying the calculations to obtain the differential entropy and the change in mutual information and therefore increasing the utility of the approach. Applications to speech processing and coding are given and potential applications to seismic signal processing, EEG classification, and ECG classification are described. Keywords: autoregressive models; entropy rate power; mutual information 1. Introduction In time series analysis, the autoregressive (AR) model, also called the linear prediction model, has received considerable attention, with a host of results on fitting AR models, AR model prediction performance, and decision-making using time series analysis based on AR models [1–3]. The linear prediction or AR model also plays a major role in many digital signal processing applications, such as linear prediction modeling of speech signals [4–6], geophysical exploration [7,8], electrocardiogram (ECG) classification [9], and electroencephalogram (EEG) classification [10,11]. -
GAIT: a Geometric Approach to Information Theory
GAIT: A Geometric Approach to Information Theory Jose Gallego Ankit Vani Max Schwarzer Simon Lacoste-Julieny Mila and DIRO, Université de Montréal Abstract supports. Optimal transport distances, such as Wasser- stein, have emerged as practical alternatives with theo- retical grounding. These methods have been used to We advocate the use of a notion of entropy compute barycenters (Cuturi and Doucet, 2014) and that reflects the relative abundances of the train generative models (Genevay et al., 2018). How- symbols in an alphabet, as well as the sim- ever, optimal transport is computationally expensive ilarities between them. This concept was as it generally lacks closed-form solutions and requires originally introduced in theoretical ecology to the solution of linear programs or the execution of study the diversity of ecosystems. Based on matrix scaling algorithms, even when solved only in this notion of entropy, we introduce geometry- approximate form (Cuturi, 2013). Approaches based aware counterparts for several concepts and on kernel methods (Gretton et al., 2012; Li et al., 2017; theorems in information theory. Notably, our Salimans et al., 2018), which take a functional analytic proposed divergence exhibits performance on view on the problem, have also been widely applied. par with state-of-the-art methods based on However, further exploration on the interplay between the Wasserstein distance, but enjoys a closed- kernel methods and information theory is lacking. form expression that can be computed effi- ciently. We demonstrate the versatility of our Contributions. We i) introduce to the machine learn- method via experiments on a broad range of ing community a similarity-sensitive definition of en- domains: training generative models, comput- tropy developed by Leinster and Cobbold(2012). -
Source Coding: Part I of Fundamentals of Source and Video Coding
Foundations and Trends R in sample Vol. 1, No 1 (2011) 1–217 c 2011 Thomas Wiegand and Heiko Schwarz DOI: xxxxxx Source Coding: Part I of Fundamentals of Source and Video Coding Thomas Wiegand1 and Heiko Schwarz2 1 Berlin Institute of Technology and Fraunhofer Institute for Telecommunica- tions — Heinrich Hertz Institute, Germany, [email protected] 2 Fraunhofer Institute for Telecommunications — Heinrich Hertz Institute, Germany, [email protected] Abstract Digital media technologies have become an integral part of the way we create, communicate, and consume information. At the core of these technologies are source coding methods that are described in this text. Based on the fundamentals of information and rate distortion theory, the most relevant techniques used in source coding algorithms are de- scribed: entropy coding, quantization as well as predictive and trans- form coding. The emphasis is put onto algorithms that are also used in video coding, which will be described in the other text of this two-part monograph. To our families Contents 1 Introduction 1 1.1 The Communication Problem 3 1.2 Scope and Overview of the Text 4 1.3 The Source Coding Principle 5 2 Random Processes 7 2.1 Probability 8 2.2 Random Variables 9 2.2.1 Continuous Random Variables 10 2.2.2 Discrete Random Variables 11 2.2.3 Expectation 13 2.3 Random Processes 14 2.3.1 Markov Processes 16 2.3.2 Gaussian Processes 18 2.3.3 Gauss-Markov Processes 18 2.4 Summary of Random Processes 19 i ii Contents 3 Lossless Source Coding 20 3.1 Classification -
Information Theory, Pattern Recognition and Neural Networks Part III Physics Exams 2006
Part III Physics exams 2004–2006 Information Theory, Pattern Recognition and Neural Networks Part III Physics exams 2006 1 A channel has a 3-bit input, x 000, 001, 010, 011, 100, 101, 110, 111 , ∈{ } and a 2-bit output y 00, 01, 10, 11 . Given an input x, the output y is ∈{ } generated by deleting exactly one of the three input bits, selected at random. For example, if the input is x = 010 then P (y x) is 1/3 for each of the outputs 00, 10, and 01; If the input is x = 001 then P (y=|01 x)=2/3 and P (y=00 x)=1/3. | | Write down the conditional entropies H(Y x=000), H(Y x=010), and H(Y x=001). | | [3] |Assuming an input distribution of the form x 000 001 010 011 100 101 110 111 1 p p p p p 1 p P (x) − 0 0 − , 2 4 4 4 4 2 work out the conditional entropy H(Y X) and show that | 2 H(Y )=1+ H p , 2 3 where H2(x)= x log2(1/x)+(1 x)log2(1/(1 x)). [3] Sketch H(Y ) and H(Y X−) as a function− of p (0, 1) on a single diagram. [5] | ∈ Sketch the mutual information I(X; Y ) as a function of p. [2] H2(1/3) 0.92. ≃ Another channel with a 3-bit input x 000, 001, 010, 011, 100, 101, 110, 111 , ∈{ } erases exactly one of its three input bits, marking the erased symbol by a ?. -
On a General Definition of Conditional Rényi Entropies
proceedings Proceedings On a General Definition of Conditional Rényi Entropies † Velimir M. Ili´c 1,*, Ivan B. Djordjevi´c 2 and Miomir Stankovi´c 3 1 Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Beograd, Serbia 2 Department of Electrical and Computer Engineering, University of Arizona, 1230 E. Speedway Blvd, Tucson, AZ 85721, USA; [email protected] 3 Faculty of Occupational Safety, University of Niš, Carnojevi´ca10a,ˇ 18000 Niš, Serbia; [email protected] * Correspondence: [email protected] † Presented at the 4th International Electronic Conference on Entropy and Its Applications, 21 November–1 December 2017; Available online: http://sciforum.net/conference/ecea-4. Published: 21 November 2017 Abstract: In recent decades, different definitions of conditional Rényi entropy (CRE) have been introduced. Thus, Arimoto proposed a definition that found an application in information theory, Jizba and Arimitsu proposed a definition that found an application in time series analysis and Renner-Wolf, Hayashi and Cachin proposed definitions that are suitable for cryptographic applications. However, there is still no a commonly accepted definition, nor a general treatment of the CRE-s, which can essentially and intuitively be represented as an average uncertainty about a random variable X if a random variable Y is given. In this paper we fill the gap and propose a three-parameter CRE, which contains all of the previous definitions as special cases that can be obtained by a proper choice of the parameters. Moreover, it satisfies all of the properties that are simultaneously satisfied by the previous definitions, so that it can successfully be used in aforementioned applications.