DOCSLIB.ORG

10.3 Joint Differential Entropy, Conditional (Differential) Entropy, and Mutual Information

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
10.3 Joint Differential Entropy, Conditional (Differential) Entropy, and Mutual Information 10.3 Joint Differential Entropy, Conditional (Differential) Entropy, and Mutual Information Definition 10.17 The joint di↵erential entropy h(X) of a random vector X with joint pdf f(x) is defined as h(X)= f(x) log f(x)dx = E log f(X). − − ZS Corollary If X ,X , ,X are mutually independent, then 1 2 ··· n n h(X)= h(Xi). i=1 X Theorem 10.18 (Translation) h(X + c)=h(X). Theorem 10.19 (Scaling) h(AX)=h(X) + log det(A) . | | Definition 10.17 The joint di↵erential entropy h(X) of a random vector X with joint pdf f(x) is defined as h(X)= f(x) log f(x)dx = E log f(X). − − ZS Corollary If X ,X , ,X are mutually independent, then 1 2 ··· n n h(X)= h(Xi). i=1 X Theorem 10.18 (Translation) h(X + c)=h(X). Theorem 10.19 (Scaling) h(AX)=h(X) + log det(A) . | | Definition 10.17 The joint di↵erential entropy h(X) of a random vector X with joint pdf f(x) is defined as h(X)= f(x) log f(x)dx = E log f(X). − − ZS Corollary If X ,X , ,X are mutually independent, then 1 2 ··· n n h(X)= h(Xi). i=1 X Theorem 10.18 (Translation) h(X + c)=h(X). Theorem 10.19 (Scaling) h(AX)=h(X) + log det(A) . | | Definition 10.17 The joint di↵erential entropy h(X) of a random vector X with joint pdf f(x) is defined as h(X)= f(x) log f(x)dx = E log f(X). − − ZS Corollary If X ,X , ,X are mutually independent, then 1 2 ··· n n h(X)= h(Xi). i=1 X Theorem 10.18 (Translation) h(X + c)=h(X). Theorem 10.19 (Scaling) h(AX)=h(X) + log det(A) . | | Theorem 10.20 (Multivariate Gaussian Distribution) Let X (µ,K). Then ∼ N 1 h(X)= log [(2πe)n K ] . 2 | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 det(Q)= 1 ± n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | K = Q⇤Q> K = Q ⇤ Q> | | | || || | 2 = Q ⇤ | | | | = ⇤ | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 det(Q)= 1 ± n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | K = Q⇤Q> K = Q ⇤ Q> | | | || || | 2 = Q ⇤ | | | | = ⇤ | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 det(Q)= 1 ± n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | K = Q⇤Q> K = Q ⇤ Q> | | | || || | 2 = Q ⇤ | | | | = ⇤ | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 det(Q)= 1 ± n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | K = Q⇤Q> K = Q ⇤ Q> | | | || || | 2 = Q ⇤ | | | | = ⇤ | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 det(Q)= 1 ± n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | K = Q⇤Q> K = Q ⇤ Q> | | | || || | 2 = Q ⇤ | | | | = ⇤ | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated. iY=1 4 5 1 n = log[(2⇡e) ⇤ ] 2 | | 1 n = log[(2⇡e) K ]. 2 | | Theorem 10.20 (Multivariate Gaussian Distribu- 5. Now consider tion) Let X (µ,K). Then ⇠N h(X)=h(QY) 1 n = h(Y)+log det(Q) Theorem 10.19 h(X)= log (2⇡e) K . | | 2 | | h i = h(Y)+0 n Proof = h(Yi) 1. Let K be diagonalizable as Q⇤Q>. iX=1 2. Write X = QY, where the random variables in Y n 1 are uncorrelated with var Yi = λi, the ith diagonal = log(2⇡eλi) 2 element of ⇤(cf. Corollary 10.7). iX=1 3. Since X is Gaussian, so is Y. n 1 n = log (2⇡e) λi 4. Then the random variables in Y are mutually inde- 2 2 3 pendent because they are uncorrelated.
Load more

Recommended publications
  • Distribution of Mutual Information
    Distribution of Mutual Information Marcus Hutter IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland [email protected] http://www.idsia.ch/-marcus Abstract The mutual information of two random variables z and J with joint probabilities {7rij} is commonly used in learning Bayesian nets as well as in many other fields. The chances 7rij are usually estimated by the empirical sampling frequency nij In leading to a point es­ timate J(nij In) for the mutual information. To answer questions like "is J (nij In) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point es­ timate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p( 7r) comprising prior information about 7r. From the prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln) of the mutual information can be cal­ culated. We derive reliable and quickly computable approximations for p(Iln). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed. 1 Introduction The mutual information J (also called cross entropy) is a widely used information theoretic measure for the stochastic dependency of random variables [CT91, SooOO] . It is used, for instance, in learning Bayesian nets [Bun96, Hec98] , where stochasti­ cally dependent nodes shall be connected. The mutual information defined in (1) can be computed if the joint probabilities {7rij} of the two random variables z and J are known.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 .
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • A Statistical Framework for Neuroimaging Data Analysis Based on Mutual Information Estimated Via a Gaussian Copula
    bioRxiv preprint doi: https://doi.org/10.1101/043745; this version posted October 25, 2016. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. A statistical framework for neuroimaging data analysis based on mutual information estimated via a Gaussian copula Robin A. A. Ince1*, Bruno L. Giordano1, Christoph Kayser1, Guillaume A. Rousselet1, Joachim Gross1 and Philippe G. Schyns1 1 Institute of Neuroscience and Psychology, University of Glasgow, 58 Hillhead Street, Glasgow, G12 8QB, UK * Corresponding author: [email protected] +44 7939 203 596 Abstract We begin by reviewing the statistical framework of information theory as applicable to neuroimaging data analysis. A major factor hindering wider adoption of this framework in neuroimaging is the difficulty of estimating information theoretic quantities in practice. We present a novel estimation technique that combines the statistical theory of copulas with the closed form solution for the entropy of Gaussian variables. This results in a general, computationally efficient, flexible, and robust multivariate statistical framework that provides effect sizes on a common meaningful scale, allows for unified treatment of discrete, continuous, uni- and multi-dimensional variables, and enables direct comparisons of representations from behavioral and brain responses across any recording modality. We validate the use of this estimate as a statistical test within a neuroimaging context, considering both discrete stimulus classes and continuous stimulus features. We also present examples of analyses facilitated by these developments, including application of multivariate analyses to MEG planar magnetic field gradients, and pairwise temporal interactions in evoked EEG responses.
    [Show full text]
  • Lecture 11: Channel Coding Theorem: Converse Part 1 Recap
    EE376A/STATS376A Information Theory Lecture 11 - 02/13/2018 Lecture 11: Channel Coding Theorem: Converse Part Lecturer: Tsachy Weissman Scribe: Erdem Bıyık In this lecture1, we will continue our discussion on channel coding theory. In the previous lecture, we proved the direct part of the theorem, which suggests if R < C(I), then R is achievable. Now, we are going to prove the converse statement: If R > C(I), then R is not achievable. We will also state some important notes about the direct and converse parts. 1 Recap: Communication Problem Recall the communication problem: Memoryless Channel n n J ∼ Uniff1; 2;:::;Mg −! encoder −!X P (Y jX) −!Y decoder −! J^ log M bits • Rate = R = n channel use • Probability of error = Pe = P (J^ 6= J) (I) (I) The main result is C = C = maxPX I(X; Y ). Last week, we showed R is achievable if R < C . In this lecture, we are going to prove that if R > C(I), then R is not achievable. 2 Fano's Inequality Theorem (Fano's Inequality). Let X be a discrete random variable and X^ = X^(Y ) be a guess of X based on Y . Let Pe = P (X 6= X^). Then, H(XjY ) ≤ h2(Pe) + Pe log(jX j − 1) where h2 is the binary entropy function. ^ Proof. Let V = 1fX6=X^ g, i.e. V is 1 if X 6= X and 0 otherwise. H(XjY ) ≤ H(X; V jY ) (1) = H(V jY ) + H(XjV; Y ) (2) ≤ H(V ) + H(XjV; Y ) (3) X = H(V ) + H(XjV =v; Y =y)P (V =v; Y =y) (4) v;y X X = H(V ) + H(XjV =0;Y =y)P (V =0;Y =y) + H(XjV =1;Y =y)P (V =1;Y =y) (5) y y X = H(V ) + H(XjV =1;Y =y)P (V =1;Y =y) (6) y X ≤ H(V ) + log(jX j − 1) P (V =1;Y =y) (7) y = H(V ) + log(jX j − 1)P (V =1) (8) = h2(Pe) + Pe log(jX j − 1) (9) 1Reading: Chapter 7.9 and 7.12 of Cover, Thomas M., and Joy A.
    [Show full text]
  • Information Theory 1 Entropy 2 Mutual Information
    CS769 Spring 2010 Advanced Natural Language Processing Information Theory Lecturer: Xiaojin Zhu [email protected] In this lecture we will learn about entropy, mutual information, KL-divergence, etc., which are useful concepts for information processing systems. 1 Entropy Entropy of a discrete distribution p(x) over the event space X is X H(p) = − p(x) log p(x). (1) x∈X When the log has base 2, entropy has unit bits. Properties: H(p) ≥ 0, with equality only if p is deterministic (use the fact 0 log 0 = 0). Entropy is the average number of 0/1 questions needed to describe an outcome from p(x) (the Twenty Questions game). Entropy is a concave function of p. 1 1 1 1 7 For example, let X = {x1, x2, x3, x4} and p(x1) = 2 , p(x2) = 4 , p(x3) = 8 , p(x4) = 8 . H(p) = 4 bits. This definition naturally extends to joint distributions. Assuming (x, y) ∼ p(x, y), X X H(p) = − p(x, y) log p(x, y). (2) x∈X y∈Y We sometimes write H(X) instead of H(p) with the understanding that p is the underlying distribution. The conditional entropy H(Y |X) is the amount of information needed to determine Y , if the other party knows X. X X X H(Y |X) = p(x)H(Y |X = x) = − p(x, y) log p(y|x). (3) x∈X x∈X y∈Y From above, we can derive the chain rule for entropy: H(X1:n) = H(X1) + H(X2|X1) + ..
    [Show full text]
  • 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.
    [Show full text]
  • Fundamentals of Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Independent Vector Analysis (IVA)
    Advanced Signal Processing Group Fundamentals of Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Independent Vector Analysis (IVA) Dr Mohsen Naqvi School of Electrical and Electronic Engineering, Newcastle University [email protected] UDRC Summer School Advanced Signal Processing Group Acknowledgement and Recommended text: Hyvarinen et al., Independent Component Analysis, Wiley, 2001 UDRC Summer School 2 Advanced Signal Processing Group Source Separation and Component Analysis Mixing Process Un-mixing Process 푥 푠1 1 푦1 H W Independent? 푠푛 푥푚 푦푛 Unknown Known Optimize Mixing Model: x = Hs Un-mixing Model: y = Wx = WHs =PDs UDRC Summer School 3 Advanced Signal Processing Group Principal Component Analysis (PCA) Orthogonal projection of data onto lower-dimension linear space: i. maximizes variance of projected data ii. minimizes mean squared distance between data point and projections UDRC Summer School 4 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 2-D Gaussian Data UDRC Summer School 5 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 1st PC in the direction of the largest variance UDRC Summer School 6 Advanced Signal Processing Group Principal Component Analysis (PCA) Example: 2nd PC in the direction of the second largest variance and each PC is orthogonal to other components. UDRC Summer School 7 Advanced Signal Processing Group Principal Component Analysis (PCA) • In PCA the redundancy is measured by correlation between data elements • Using
    [Show full text]