Neural coding Ecological approach to sensory coding: efficient adaptation to the natural environment

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Neural coding Ecological approach to sensory coding: efficient adaptation to the natural environment"

Transcription

1 Neural coding Ecological approach to sensory coding: efficient adaptation to the natural environment Jean-Pierre Nadal CNRS & EHESS Laboratoire de Physique Statistique (LPS, UMR 8550 CNRS - ENS UPMC Univ. Paris Diderot) Ecole Normale Supérieure (ENS) & Centre d Analyse et de Mathématique Sociales (CAMS, UMR 8557 CNRS - EHESS) Ecole des Hautes Etudes en Sciences Sociales (EHESS)

2 Neural coding Photoreceptors intensities data Retina Principal Component Analysis (PCA) Neural representation: activivities of ganglion cells Representation: projection onto the principal axis environment stimulus network neural representation data algorithm (neural code) signal filter ρ (.) θ X θ = { θ 1, θ 2,, θ N } X = { X 1, X 2,, X p }

3 Ecological approach to sensory coding: efficient adaptation to the natural environment Horace Barlow, 1961 H. B. Barlow. Possible principles underlying the transformation of sensory messages. Sensory Communication, pp , 1961 efficient coding hypothesis sensory processing in the brain should be adapted to natural stimuli e.g. neurons in the visual (or auditory) system of a given animal should be optimized for coding images (or sounds) representative of those found in the natural environment of that animal. It has been shown that filters optimized for coding natural images lead to filters which resemble the receptive fields of simple-cells in V1. In the auditory domain, optimizing a network for coding natural sounds leads to filters which resemble the impulse response of cochlear filters found in the inner ear. Formalization: tools from Information Theory, Statistical (Bayesian) inference, parameter estimation

4 Neural coding PCA: max variances, well adapted to Gaussian like distributions More general: «infomax» Max Mutual Information[stimuli ; neural representation] environment stimulus network neural representation data algorithm (neural code) signal filter ρ (.) θ W X θ = { θ 1, θ 2,, θ N } X = { X 1, X 2,, X p }

5 Information Theory (Shannon) Entropy, Shannon information Mutual Information = output entropy equivocation o Capacity o Infomax Redundancy o different types of redundancies o min redundancy (H. Barlow, 1961) = Independent Component Analysis (ICA) Parameter estimation Cramer-Rao inequality Fisher Information

6 Entropy (Shannon information) Basic properties Discrete case Σ k p k = 1 (k=1,, K) H = - Σ k p k ln p k ln K Max entropy: equiprobable distribution, p k = 1/K H = ln K Binary case: 1 bit of information = 1 binary variable with equiprobable states (fair coin) H = ln 2 H / ln 2 = 1 (bit) With the logarithm in base 2: information in bits

7 Binary case binary random variable taking one value with probability f and the other with probability 1-f H = f ln f 1 f ln 1 f With logarithms in base 2: information in bits H 2 = f log 2 f (1 f) log 2 (1 f) log 2 (. ) = ln(. ) ln 2 H 2 1 H 2 f = H 2 (1 f) H 2 f = 0 = H 2 (f = 1) = /2 1 f For f = 1 2 H 2 = 1 bit

8 Entropy (Shannon information) Continuous case x X; probability distribution differential entropy H =

9 Entropy (Shannon information) Basic properties Max entropy? Continuous case (differential) entropy H = - ρ (x) ln ρ (x) dx among distributions ρ with support [a, b]: uniform distribution on [a, b]; ρ =1/(b-a); H = ln (b-a) among distributions ρ with support ]-, [ ρ (x) = under variance constraint, <x 2 >= σ 2 Gaussian distribution; H = ½ ln(2 π e σ 2 )

10 Entropy (Shannon information): simple examples Uniform distribution Gaussian distribution Multidimensional Gaussian distribution

11 Mutual information environment stimulus neural representation ρ (.) θ Q( X θ ) X = { X 1, X 2,, X p } I [ θ, X ] = [ entropy of X ] [ entropy of X given θ ] (equivocation) = ln Q X Q X d p X + ln Q(X θ) Q(X θ) d p X ρ θ d N θ = Information that X carries about θ = Information that θ carries about X = H X + H θ H(θ, X) P θ, X = ln ρ θ Q X P θ, X dp X d N θ Kullback-Leibler divergence between the joint distribution and the product distribution Output distribution (marginal distribution of X): Q X = Q(X θ) ρ θ d N θ Joint distribution of X and θ : P θ, X = Q(X θ) ρ θ

12 Mutual Information = difference of entropies large number of p objects Object type: τ {, } f = probability to have an object of type Classification Data analysis Signal processing Encoding If no error Maxwell s demon H 1 = 0 H 2 = 0 Entropy (Shannon information): H H = p f ln f 1 f ln(1 f) H = Information gain = decrease in entropy I = H - H 1 - H 2 = H Box number: σ { 1, 2}

13 Mutual Information = difference of entropies large number of p objects Object type: τ {, } f = probability to have an object of type Classification Data analysis Signal processing Encoding If noise, errors Drunk Maxwell s demon H 1 > 0 H 2 > 0 Box number: σ { 1, 2} Entropy (Shannon information): H H = p f ln f 1 f ln(1 f) H = Information gain = decrease in entropy I = H - H 1 - H 2 = mutual information between τ and σ

14 Basic properties of the mutual information For any random variables, X and Y: I(X,Y) 0 I(X,Y) = 0 iff the two random variables are statistically independent Mutual info = relative entropy (Kullback-Leibler divergence) between the joint and the factorized distributions Case X discrete: I(X,Y) H(X) ln K (similarly, if Y discrete, I(X,Y) H(Y) ln M ) Data processing theorem S X Y Z I(S,Z), I(S,Y), I(X,Y) I(S,Z) I(S,Y) I(X,Y) I(X,Y) = I(S,Y) + I(X,Y S) I(S,Y)

15 Mutual information environment stimulus neural representation ρ (.) θ Q( X θ ) X = { X 1, X 2,, X p } I [ θ, X ] 0 ( I = 0 θ and X are statistically independent ) Capacity: Q given C = max I [ θ, X ρ ] (transmission channel) principle for optimal coding: Infomax ρ given max I [ θ, X ] Q Redundancy (environment) R = I [ X 1, X 2,, X p ] 0 Barlow s principle: R = Σ k I [ θ, X k min R ] - I [ θ, X ] (can be < 0)

16 Shannon: Communication theory Channel codeword of length decoding: message = = largest number of codewords that can be decoded with a fraction of error Capacity: Memory less channel: ρ = proba. dist. of τ

17 Stimulus S output V Mutual info I (V,S) = output entropy - equivocation equivocation = entropy of the output given the stimulus, averaged over the stimulus distribution A useful particular case: additive noise V = f(s) + noise equivocation? = noise entropy (hence independent of the input distribution) I (V,S) = output entropy - noise entropy

Introduction to Information Theory

Introduction to Information Theory Introduction to Information Theory Gurinder Singh Mickey Atwal atwal@cshl.edu Center for Quantitative Biology Kullback-Leibler Divergence Summary Shannon s coding theorems Entropy Mutual Information Multi-information

More information

Information in Biology

Information in Biology Information in Biology CRI - Centre de Recherches Interdisciplinaires, Paris May 2012 Information processing is an essential part of Life. Thinking about it in quantitative terms may is useful. 1 Living

More information

Information Theory. Coding and Information Theory. Information Theory Textbooks. Entropy

Information Theory. Coding and Information Theory. Information Theory Textbooks. Entropy Coding and Information Theory Chris Williams, School of Informatics, University of Edinburgh Overview What is information theory? Entropy Coding Information Theory Shannon (1948): Information theory is

More information

Entropies & Information Theory

Entropies & Information Theory Entropies & Information Theory LECTURE I Nilanjana Datta University of Cambridge,U.K. See lecture notes on: http://www.qi.damtp.cam.ac.uk/node/223 Quantum Information Theory Born out of Classical Information

More information

Introduction to Information Theory. Uncertainty. Entropy. Surprisal. Joint entropy. Conditional entropy. Mutual information.

Introduction to Information Theory. Uncertainty. Entropy. Surprisal. Joint entropy. Conditional entropy. Mutual information. L65 Dept. of Linguistics, Indiana University Fall 205 Information theory answers two fundamental questions in communication theory: What is the ultimate data compression? What is the transmission rate

More information

Dept. of Linguistics, Indiana University Fall 2015

Dept. of Linguistics, Indiana University Fall 2015 L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 28 Information theory answers two fundamental questions in communication theory: What is the ultimate data compression? What is the transmission

More information

Machine Learning Srihari. Information Theory. Sargur N. Srihari

Machine Learning Srihari. Information Theory. Sargur N. Srihari Information Theory Sargur N. Srihari 1 Topics 1. Entropy as an Information Measure 1. Discrete variable definition Relationship to Code Length 2. Continuous Variable Differential Entropy 2. Maximum Entropy

More information

Population Coding. Maneesh Sahani Gatsby Computational Neuroscience Unit University College London

Population Coding. Maneesh Sahani Gatsby Computational Neuroscience Unit University College London Population Coding Maneesh Sahani maneesh@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit University College London Term 1, Autumn 2010 Coding so far... Time-series for both spikes and stimuli Empirical

More information

Chapter 9 Fundamental Limits in Information Theory

Chapter 9 Fundamental Limits in Information Theory Chapter 9 Fundamental Limits in Information Theory Information Theory is the fundamental theory behind information manipulation, including data compression and data transmission. 9.1 Introduction o For

More information

Revision of Lecture 4

Revision of Lecture 4 Revision of Lecture 4 We have completed studying digital sources from information theory viewpoint We have learnt all fundamental principles for source coding, provided by information theory Practical

More information

One Lesson of Information Theory

One Lesson of Information Theory Institut für One Lesson of Information Theory Prof. Dr.-Ing. Volker Kühn Institute of Communications Engineering University of Rostock, Germany Email: volker.kuehn@uni-rostock.de http://www.int.uni-rostock.de/

More information

3F1: Signals and Systems INFORMATION THEORY Examples Paper Solutions

3F1: Signals and Systems INFORMATION THEORY Examples Paper Solutions Engineering Tripos Part IIA THIRD YEAR 3F: Signals and Systems INFORMATION THEORY Examples Paper Solutions. Let the joint probability mass function of two binary random variables X and Y be given in the

More information

Capacity of AWGN channels

Capacity of AWGN channels Chapter 3 Capacity of AWGN channels In this chapter we prove that the capacity of an AWGN channel with bandwidth W and signal-tonoise ratio SNR is W log 2 (1+SNR) bits per second (b/s). The proof that

More information

Information Theory, Statistics, and Decision Trees

Information Theory, Statistics, and Decision Trees Information Theory, Statistics, and Decision Trees Léon Bottou COS 424 4/6/2010 Summary 1. Basic information theory. 2. Decision trees. 3. Information theory and statistics. Léon Bottou 2/31 COS 424 4/6/2010

More information

Information Theory. David Rosenberg. June 15, New York University. David Rosenberg (New York University) DS-GA 1003 June 15, / 18

Information Theory. David Rosenberg. June 15, New York University. David Rosenberg (New York University) DS-GA 1003 June 15, / 18 Information Theory David Rosenberg New York University June 15, 2015 David Rosenberg (New York University) DS-GA 1003 June 15, 2015 1 / 18 A Measure of Information? Consider a discrete random variable

More information

Block 2: Introduction to Information Theory

Block 2: Introduction to Information Theory Block 2: Introduction to Information Theory Francisco J. Escribano April 26, 2015 Francisco J. Escribano Block 2: Introduction to Information Theory April 26, 2015 1 / 51 Table of contents 1 Motivation

More information

Lateral organization & computation

Lateral organization & computation Lateral organization & computation review Population encoding & decoding lateral organization Efficient representations that reduce or exploit redundancy Fixation task 1rst order Retinotopic maps Log-polar

More information

Tutorial on Blind Source Separation and Independent Component Analysis

Tutorial on Blind Source Separation and Independent Component Analysis Tutorial on Blind Source Separation and Independent Component Analysis Lucas Parra Adaptive Image & Signal Processing Group Sarnoff Corporation February 09, 2002 Linear Mixtures... problem statement...

More information

Multimedia Communications. Scalar Quantization

Multimedia Communications. Scalar Quantization Multimedia Communications Scalar Quantization Scalar Quantization In many lossy compression applications we want to represent source outputs using a small number of code words. Process of representing

More information

Pacific Symposium on Biocomputing 6: (2001)

Pacific Symposium on Biocomputing 6: (2001) Analyzing sensory systems with the information distortion function Alexander G Dimitrov and John P Miller Center for Computational Biology Montana State University Bozeman, MT 59715-3505 falex,jpmg@nervana.montana.edu

More information

EE/Stat 376B Handout #5 Network Information Theory October, 14, Homework Set #2 Solutions

EE/Stat 376B Handout #5 Network Information Theory October, 14, Homework Set #2 Solutions EE/Stat 376B Handout #5 Network Information Theory October, 14, 014 1. Problem.4 parts (b) and (c). Homework Set # Solutions (b) Consider h(x + Y ) h(x + Y Y ) = h(x Y ) = h(x). (c) Let ay = Y 1 + Y, where

More information

CS 630 Basic Probability and Information Theory. Tim Campbell

CS 630 Basic Probability and Information Theory. Tim Campbell CS 630 Basic Probability and Information Theory Tim Campbell 21 January 2003 Probability Theory Probability Theory is the study of how best to predict outcomes of events. An experiment (or trial or event)

More information

Coding for Discrete Source

Coding for Discrete Source EGR 544 Communication Theory 3. Coding for Discrete Sources Z. Aliyazicioglu Electrical and Computer Engineering Department Cal Poly Pomona Coding for Discrete Source Coding Represent source data effectively

More information

Information maximization in a network of linear neurons

Information maximization in a network of linear neurons Information maximization in a network of linear neurons Holger Arnold May 30, 005 1 Introduction It is known since the work of Hubel and Wiesel [3], that many cells in the early visual areas of mammals

More information

An Introduction to Independent Components Analysis (ICA)

An Introduction to Independent Components Analysis (ICA) An Introduction to Independent Components Analysis (ICA) Anish R. Shah, CFA Northfield Information Services Anish@northinfo.com Newport Jun 6, 2008 1 Overview of Talk Review principal components Introduce

More information

An instantaneous code (prefix code, tree code) with the codeword lengths l 1,..., l N exists if and only if. 2 l i. i=1

An instantaneous code (prefix code, tree code) with the codeword lengths l 1,..., l N exists if and only if. 2 l i. i=1 Kraft s inequality An instantaneous code (prefix code, tree code) with the codeword lengths l 1,..., l N exists if and only if N 2 l i 1 Proof: Suppose that we have a tree code. Let l max = max{l 1,...,

More information

Information Theory CHAPTER. 5.1 Introduction. 5.2 Entropy

Information Theory CHAPTER. 5.1 Introduction. 5.2 Entropy Haykin_ch05_pp3.fm Page 207 Monday, November 26, 202 2:44 PM CHAPTER 5 Information Theory 5. Introduction As mentioned in Chapter and reiterated along the way, the purpose of a communication system is

More information

MAHALAKSHMI ENGINEERING COLLEGE-TRICHY QUESTION BANK UNIT V PART-A. 1. What is binary symmetric channel (AUC DEC 2006)

MAHALAKSHMI ENGINEERING COLLEGE-TRICHY QUESTION BANK UNIT V PART-A. 1. What is binary symmetric channel (AUC DEC 2006) MAHALAKSHMI ENGINEERING COLLEGE-TRICHY QUESTION BANK SATELLITE COMMUNICATION DEPT./SEM.:ECE/VIII UNIT V PART-A 1. What is binary symmetric channel (AUC DEC 2006) 2. Define information rate? (AUC DEC 2007)

More information

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road UNIT I

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road UNIT I SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : CODING THEORY & TECHNIQUES(16EC3810) Course & Branch: M.Tech - DECS

More information

Chapter I: Fundamental Information Theory

Chapter I: Fundamental Information Theory ECE-S622/T62 Notes Chapter I: Fundamental Information Theory Ruifeng Zhang Dept. of Electrical & Computer Eng. Drexel University. Information Source Information is the outcome of some physical processes.

More information

Foundations of Statistical Inference

Foundations of Statistical Inference Foundations of Statistical Inference Julien Berestycki Department of Statistics University of Oxford MT 2016 Julien Berestycki (University of Oxford) SB2a MT 2016 1 / 32 Lecture 14 : Variational Bayes

More information

Introduction to Convolutional Codes, Part 1

Introduction to Convolutional Codes, Part 1 Introduction to Convolutional Codes, Part 1 Frans M.J. Willems, Eindhoven University of Technology September 29, 2009 Elias, Father of Coding Theory Textbook Encoder Encoder Properties Systematic Codes

More information

ECE598: Information-theoretic methods in high-dimensional statistics Spring 2016

ECE598: Information-theoretic methods in high-dimensional statistics Spring 2016 ECE598: Information-theoretic methods in high-dimensional statistics Spring 06 Lecture : Mutual Information Method Lecturer: Yihong Wu Scribe: Jaeho Lee, Mar, 06 Ed. Mar 9 Quick review: Assouad s lemma

More information

Exercise 1. = P(y a 1)P(a 1 )

Exercise 1. = P(y a 1)P(a 1 ) Chapter 7 Channel Capacity Exercise 1 A source produces independent, equally probable symbols from an alphabet {a 1, a 2 } at a rate of one symbol every 3 seconds. These symbols are transmitted over a

More information

X 1 : X Table 1: Y = X X 2

X 1 : X Table 1: Y = X X 2 ECE 534: Elements of Information Theory, Fall 200 Homework 3 Solutions (ALL DUE to Kenneth S. Palacio Baus) December, 200. Problem 5.20. Multiple access (a) Find the capacity region for the multiple-access

More information

The binary entropy function

The binary entropy function ECE 7680 Lecture 2 Definitions and Basic Facts Objective: To learn a bunch of definitions about entropy and information measures that will be useful through the quarter, and to present some simple but

More information

Information Dynamics Foundations and Applications

Information Dynamics Foundations and Applications Gustavo Deco Bernd Schürmann Information Dynamics Foundations and Applications With 89 Illustrations Springer PREFACE vii CHAPTER 1 Introduction 1 CHAPTER 2 Dynamical Systems: An Overview 7 2.1 Deterministic

More information

Coding of memoryless sources 1/35

Coding of memoryless sources 1/35 Coding of memoryless sources 1/35 Outline 1. Morse coding ; 2. Definitions : encoding, encoding efficiency ; 3. fixed length codes, encoding integers ; 4. prefix condition ; 5. Kraft and Mac Millan theorems

More information

16.36 Communication Systems Engineering

16.36 Communication Systems Engineering MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.36: Communication

More information

Digital Image Processing Lectures 25 & 26

Digital Image Processing Lectures 25 & 26 Lectures 25 & 26, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2015 Area 4: Image Encoding and Compression Goal: To exploit the redundancies in the image

More information

x log x, which is strictly convex, and use Jensen s Inequality:

x log x, which is strictly convex, and use Jensen s Inequality: 2. Information measures: mutual information 2.1 Divergence: main inequality Theorem 2.1 (Information Inequality). D(P Q) 0 ; D(P Q) = 0 iff P = Q Proof. Let ϕ(x) x log x, which is strictly convex, and

More information

Information Theory - Entropy. Figure 3

Information Theory - Entropy. Figure 3 Concept of Information Information Theory - Entropy Figure 3 A typical binary coded digital communication system is shown in Figure 3. What is involved in the transmission of information? - The system

More information

Lecture 2. Capacity of the Gaussian channel

Lecture 2. Capacity of the Gaussian channel Spring, 207 5237S, Wireless Communications II 2. Lecture 2 Capacity of the Gaussian channel Review on basic concepts in inf. theory ( Cover&Thomas: Elements of Inf. Theory, Tse&Viswanath: Appendix B) AWGN

More information

Introduction to Information Theory. B. Škorić, Physical Aspects of Digital Security, Chapter 2

Introduction to Information Theory. B. Škorić, Physical Aspects of Digital Security, Chapter 2 Introduction to Information Theory B. Škorić, Physical Aspects of Digital Security, Chapter 2 1 Information theory What is it? - formal way of counting information bits Why do we need it? - often used

More information

Variational Principal Components

Variational Principal Components Variational Principal Components Christopher M. Bishop Microsoft Research 7 J. J. Thomson Avenue, Cambridge, CB3 0FB, U.K. cmbishop@microsoft.com http://research.microsoft.com/ cmbishop In Proceedings

More information

An Extended Fano s Inequality for the Finite Blocklength Coding

An Extended Fano s Inequality for the Finite Blocklength Coding An Extended Fano s Inequality for the Finite Bloclength Coding Yunquan Dong, Pingyi Fan {dongyq8@mails,fpy@mail}.tsinghua.edu.cn Department of Electronic Engineering, Tsinghua University, Beijing, P.R.

More information

Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts I-II

Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts I-II Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts I-II Gatsby Unit University College London 27 Feb 2017 Outline Part I: Theory of ICA Definition and difference

More information

The homogeneous Poisson process

The homogeneous Poisson process The homogeneous Poisson process during very short time interval Δt there is a fixed probability of an event (spike) occurring independent of what happened previously if r is the rate of the Poisson process,

More information

Correlation Detection and an Operational Interpretation of the Rényi Mutual Information

Correlation Detection and an Operational Interpretation of the Rényi Mutual Information Correlation Detection and an Operational Interpretation of the Rényi Mutual Information Masahito Hayashi 1, Marco Tomamichel 2 1 Graduate School of Mathematics, Nagoya University, and Centre for Quantum

More information

Extract. Data Analysis Tools

Extract. Data Analysis Tools Extract Data Analysis Tools Harjoat S. Bhamra July 8, 2017 Contents 1 Introduction 7 I Probability 13 2 Inequalities 15 2.1 Jensen s inequality................................ 17 2.1.1 Arithmetic Mean-Geometric

More information

(each row defines a probability distribution). Given n-strings x X n, y Y n we can use the absence of memory in the channel to compute

(each row defines a probability distribution). Given n-strings x X n, y Y n we can use the absence of memory in the channel to compute ENEE 739C: Advanced Topics in Signal Processing: Coding Theory Instructor: Alexander Barg Lecture 6 (draft; 9/6/03. Error exponents for Discrete Memoryless Channels http://www.enee.umd.edu/ abarg/enee739c/course.html

More information

Dimension Reduction (PCA, ICA, CCA, FLD,

Dimension Reduction (PCA, ICA, CCA, FLD, Dimension Reduction (PCA, ICA, CCA, FLD, Topic Models) Yi Zhang 10-701, Machine Learning, Spring 2011 April 6 th, 2011 Parts of the PCA slides are from previous 10-701 lectures 1 Outline Dimension reduction

More information

Classification & Information Theory Lecture #8

Classification & Information Theory Lecture #8 Classification & Information Theory Lecture #8 Introduction to Natural Language Processing CMPSCI 585, Fall 2007 University of Massachusetts Amherst Andrew McCallum Today s Main Points Automatically categorizing

More information

Lecture 10: Broadcast Channel and Superposition Coding

Lecture 10: Broadcast Channel and Superposition Coding Lecture 10: Broadcast Channel and Superposition Coding Scribed by: Zhe Yao 1 Broadcast channel M 0M 1M P{y 1 y x} M M 01 1 M M 0 The capacity of the broadcast channel depends only on the marginal conditional

More information

Variational Information Maximization in Gaussian Channels

Variational Information Maximization in Gaussian Channels Variational Information Maximization in Gaussian Channels Felix V. Agakov School of Informatics, University of Edinburgh, EH1 2QL, UK felixa@inf.ed.ac.uk, http://anc.ed.ac.uk David Barber IDIAP, Rue du

More information

Classical Information Theory Notes from the lectures by prof Suhov Trieste - june 2006

Classical Information Theory Notes from the lectures by prof Suhov Trieste - june 2006 Classical Information Theory Notes from the lectures by prof Suhov Trieste - june 2006 Fabio Grazioso... July 3, 2006 1 2 Contents 1 Lecture 1, Entropy 4 1.1 Random variable...............................

More information

LECTURE 13. Last time: Lecture outline

LECTURE 13. Last time: Lecture outline LECTURE 13 Last time: Strong coding theorem Revisiting channel and codes Bound on probability of error Error exponent Lecture outline Fano s Lemma revisited Fano s inequality for codewords Converse to

More information

Information Theory. Week 4 Compressing streams. Iain Murray,

Information Theory. Week 4 Compressing streams. Iain Murray, Information Theory http://www.inf.ed.ac.uk/teaching/courses/it/ Week 4 Compressing streams Iain Murray, 2014 School of Informatics, University of Edinburgh Jensen s inequality For convex functions: E[f(x)]

More information

Neural networks: Unsupervised learning

Neural networks: Unsupervised learning Neural networks: Unsupervised learning 1 Previously The supervised learning paradigm: given example inputs x and target outputs t learning the mapping between them the trained network is supposed to give

More information

An introduction to basic information theory. Hampus Wessman

An introduction to basic information theory. Hampus Wessman An introduction to basic information theory Hampus Wessman Abstract We give a short and simple introduction to basic information theory, by stripping away all the non-essentials. Theoretical bounds on

More information

Homework Set #2 Data Compression, Huffman code and AEP

Homework Set #2 Data Compression, Huffman code and AEP Homework Set #2 Data Compression, Huffman code and AEP 1. Huffman coding. Consider the random variable ( x1 x X = 2 x 3 x 4 x 5 x 6 x 7 0.50 0.26 0.11 0.04 0.04 0.03 0.02 (a Find a binary Huffman code

More information

Exam, Solutions

Exam, Solutions Exam, - Solutions Q Constructing a balanced sequence containing three kinds of stimuli Here we design a balanced cyclic sequence for three kinds of stimuli (labeled {,, }, in which every three-element

More information

arxiv:physics/ v1 [physics.data-an] 24 Apr 2000 Naftali Tishby, 1,2 Fernando C. Pereira, 3 and William Bialek 1

arxiv:physics/ v1 [physics.data-an] 24 Apr 2000 Naftali Tishby, 1,2 Fernando C. Pereira, 3 and William Bialek 1 The information bottleneck method arxiv:physics/0004057v1 [physics.data-an] 24 Apr 2000 Naftali Tishby, 1,2 Fernando C. Pereira, 3 and William Bialek 1 1 NEC Research Institute, 4 Independence Way Princeton,

More information

CS546:Learning and NLP Lec 4: Mathematical and Computational Paradigms

CS546:Learning and NLP Lec 4: Mathematical and Computational Paradigms CS546:Learning and NLP Lec 4: Mathematical and Computational Paradigms Spring 2009 February 3, 2009 Lecture Some note on statistics Bayesian decision theory Concentration bounds Information theory Introduction

More information

No. of dimensions 1. No. of centers

No. of dimensions 1. No. of centers Contents 8.6 Course of dimensionality............................ 15 8.7 Computational aspects of linear estimators.................. 15 8.7.1 Diagonalization of circulant andblock-circulant matrices......

More information

Independent Component Analysis and Unsupervised Learning. Jen-Tzung Chien

Independent Component Analysis and Unsupervised Learning. Jen-Tzung Chien Independent Component Analysis and Unsupervised Learning Jen-Tzung Chien TABLE OF CONTENTS 1. Independent Component Analysis 2. Case Study I: Speech Recognition Independent voices Nonparametric likelihood

More information

Mutual Information, Synergy and Some Curious Phenomena for Simple Channels

Mutual Information, Synergy and Some Curious Phenomena for Simple Channels Mutual Information, Synergy and Some Curious henomena for Simple Channels I. Kontoyiannis Div of Applied Mathematics & Dpt of Computer Science Brown University rovidence, RI 9, USA Email: yiannis@dam.brown.edu

More information

Lecture 11: Quantum Information III - Source Coding

Lecture 11: Quantum Information III - Source Coding CSCI5370 Quantum Computing November 25, 203 Lecture : Quantum Information III - Source Coding Lecturer: Shengyu Zhang Scribe: Hing Yin Tsang. Holevo s bound Suppose Alice has an information source X that

More information

Synergy, Redundancy, and Independence in Population Codes, Revisited

Synergy, Redundancy, and Independence in Population Codes, Revisited The Journal of Neuroscience, May 25, 2005 25(21):5195 5206 5195 Behavioral/Systems/Cognitive Synergy, Redundancy, and Independence in Population Codes, Revisited Peter E. Latham 1 and Sheila Nirenberg

More information

Statistical mechanics and capacity-approaching error-correctingcodes

Statistical mechanics and capacity-approaching error-correctingcodes Physica A 302 (2001) 14 21 www.elsevier.com/locate/physa Statistical mechanics and capacity-approaching error-correctingcodes Nicolas Sourlas Laboratoire de Physique Theorique de l, UMR 8549, Unite Mixte

More information

Chapter 2: Source coding

Chapter 2: Source coding Chapter 2: meghdadi@ensil.unilim.fr University of Limoges Chapter 2: Entropy of Markov Source Chapter 2: Entropy of Markov Source Markov model for information sources Given the present, the future is independent

More information

List Decoding of Reed Solomon Codes

List Decoding of Reed Solomon Codes List Decoding of Reed Solomon Codes p. 1/30 List Decoding of Reed Solomon Codes Madhu Sudan MIT CSAIL Background: Reliable Transmission of Information List Decoding of Reed Solomon Codes p. 2/30 List Decoding

More information

much more on minimax (order bounds) cf. lecture by Iain Johnstone

much more on minimax (order bounds) cf. lecture by Iain Johnstone much more on minimax (order bounds) cf. lecture by Iain Johnstone http://www-stat.stanford.edu/~imj/wald/wald1web.pdf today s lecture parametric estimation, Fisher information, Cramer-Rao lower bound:

More information

Lower Bounds on the Graphical Complexity of Finite-Length LDPC Codes

Lower Bounds on the Graphical Complexity of Finite-Length LDPC Codes Lower Bounds on the Graphical Complexity of Finite-Length LDPC Codes Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel 2009 IEEE International

More information

A Holevo-type bound for a Hilbert Schmidt distance measure

A Holevo-type bound for a Hilbert Schmidt distance measure Journal of Quantum Information Science, 205, *,** Published Online **** 204 in SciRes. http://www.scirp.org/journal/**** http://dx.doi.org/0.4236/****.204.***** A Holevo-type bound for a Hilbert Schmidt

More information

Maximum mutual information vector quantization of log-likelihood ratios for memory efficient HARQ implementations

Maximum mutual information vector quantization of log-likelihood ratios for memory efficient HARQ implementations Downloaded from orbit.dtu.dk on: Apr 29, 2018 Maximum mutual information vector quantization of log-likelihood ratios for memory efficient HARQ implementations Danieli, Matteo; Forchhammer, Søren; Andersen,

More information

Shannon's Theory of Communication

Shannon's Theory of Communication Shannon's Theory of Communication An operational introduction 5 September 2014, Introduction to Information Systems Giovanni Sileno g.sileno@uva.nl Leibniz Center for Law University of Amsterdam Fundamental

More information

(Classical) Information Theory II: Source coding

(Classical) Information Theory II: Source coding (Classical) Information Theory II: Source coding Sibasish Ghosh The Institute of Mathematical Sciences CIT Campus, Taramani, Chennai 600 113, India. p. 1 Abstract The information content of a random variable

More information

When Do Microcircuits Produce Beyond-Pairwise Correlations?

When Do Microcircuits Produce Beyond-Pairwise Correlations? When Do Microcircuits Produce Beyond-Pairwise Correlations? Andrea K. Barreiro,4,, Julijana Gjorgjieva 3,5, Fred Rieke 2, and Eric Shea-Brown Department of Applied Mathematics, University of Washington

More information

Application of Information Theory, Lecture 7. Relative Entropy. Handout Mode. Iftach Haitner. Tel Aviv University.

Application of Information Theory, Lecture 7. Relative Entropy. Handout Mode. Iftach Haitner. Tel Aviv University. Application of Information Theory, Lecture 7 Relative Entropy Handout Mode Iftach Haitner Tel Aviv University. December 1, 2015 Iftach Haitner (TAU) Application of Information Theory, Lecture 7 December

More information

Analysis of neural coding through quantization with an information-based distortion measure

Analysis of neural coding through quantization with an information-based distortion measure Submitted to: Network: Comput. Neural Syst. Analysis of neural coding through quantization with an information-based distortion measure Alexander G. Dimitrov, John P. Miller, Tomáš Gedeon, Zane Aldworth

More information

Lec 03 Entropy and Coding II Hoffman and Golomb Coding

Lec 03 Entropy and Coding II Hoffman and Golomb Coding CS/EE 5590 / ENG 40 Special Topics Multimedia Communication, Spring 207 Lec 03 Entropy and Coding II Hoffman and Golomb Coding Zhu Li Z. Li Multimedia Communciation, 207 Spring p. Outline Lecture 02 ReCap

More information

C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University

C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University Quantization C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University http://www.csie.nctu.edu.tw/~cmliu/courses/compression/ Office: EC538 (03)5731877 cmliu@cs.nctu.edu.tw

More information

Perception of the structure of the physical world using unknown multimodal sensors and effectors

Perception of the structure of the physical world using unknown multimodal sensors and effectors Perception of the structure of the physical world using unknown multimodal sensors and effectors D. Philipona Sony CSL, 6 rue Amyot 75005 Paris, France david.philipona@m4x.org J.K. O Regan Laboratoire

More information

4.2 Entropy lost and information gained

4.2 Entropy lost and information gained 4.2. ENTROPY LOST AND INFORMATION GAINED 101 4.2 Entropy lost and information gained Returning to the conversation between Max and Allan, we assumed that Max would receive a complete answer to his question,

More information

EVALUATION OF PACKET ERROR RATE IN WIRELESS NETWORKS

EVALUATION OF PACKET ERROR RATE IN WIRELESS NETWORKS EVALUATION OF PACKET ERROR RATE IN WIRELESS NETWORKS Ramin Khalili, Kavé Salamatian LIP6-CNRS, Université Pierre et Marie Curie. Paris, France. Ramin.khalili, kave.salamatian@lip6.fr Abstract Bit Error

More information

MAXIMUM ENTROPIES COPULAS

MAXIMUM ENTROPIES COPULAS MAXIMUM ENTROPIES COPULAS Doriano-Boris Pougaza & Ali Mohammad-Djafari Groupe Problèmes Inverses Laboratoire des Signaux et Systèmes (UMR 8506 CNRS - SUPELEC - UNIV PARIS SUD) Supélec, Plateau de Moulon,

More information

Patch similarity under non Gaussian noise

Patch similarity under non Gaussian noise The 18th IEEE International Conference on Image Processing Brussels, Belgium, September 11 14, 011 Patch similarity under non Gaussian noise Charles Deledalle 1, Florence Tupin 1, Loïc Denis 1 Institut

More information

1 Introduction to information theory

1 Introduction to information theory 1 Introduction to information theory 1.1 Introduction In this chapter we present some of the basic concepts of information theory. The situations we have in mind involve the exchange of information through

More information

Optimal codes - I. A code is optimal if it has the shortest codeword length L. i i. This can be seen as an optimization problem. min.

Optimal codes - I. A code is optimal if it has the shortest codeword length L. i i. This can be seen as an optimization problem. min. Huffman coding Optimal codes - I A code is optimal if it has the shortest codeword length L L m = i= pl i i This can be seen as an optimization problem min i= li subject to D m m i= lp Gabriele Monfardini

More information

Information Theory. M1 Informatique (parcours recherche et innovation) Aline Roumy. January INRIA Rennes 1/ 73

Information Theory. M1 Informatique (parcours recherche et innovation) Aline Roumy. January INRIA Rennes 1/ 73 1/ 73 Information Theory M1 Informatique (parcours recherche et innovation) Aline Roumy INRIA Rennes January 2018 Outline 2/ 73 1 Non mathematical introduction 2 Mathematical introduction: definitions

More information

Approximate Inference Part 1 of 2

Approximate Inference Part 1 of 2 Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ Bayesian paradigm Consistent use of probability theory

More information

Adaptive contrast gain control and information maximization $

Adaptive contrast gain control and information maximization $ Neurocomputing 65 66 (2005) 6 www.elsevier.com/locate/neucom Adaptive contrast gain control and information maximization $ Yuguo Yu a,, Tai Sing Lee b a Center for the Neural Basis of Cognition, Carnegie

More information

Information and Entropy

Information and Entropy Information and Entropy Shannon s Separation Principle Source Coding Principles Entropy Variable Length Codes Huffman Codes Joint Sources Arithmetic Codes Adaptive Codes Thomas Wiegand: Digital Image Communication

More information

Lecture 17: Differential Entropy

Lecture 17: Differential Entropy Lecture 17: Differential Entropy Differential entropy AEP for differential entropy Quantization Maximum differential entropy Estimation counterpart of Fano s inequality Dr. Yao Xie, ECE587, Information

More information

Approximate Inference Part 1 of 2

Approximate Inference Part 1 of 2 Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ 1 Bayesian paradigm Consistent use of probability theory

More information

The Minimum Message Length Principle for Inductive Inference

The Minimum Message Length Principle for Inductive Inference The Principle for Inductive Inference Centre for Molecular, Environmental, Genetic & Analytic (MEGA) Epidemiology School of Population Health University of Melbourne University of Helsinki, August 25,

More information

Sensory Integration and Density Estimation

Sensory Integration and Density Estimation Sensory Integration and Density Estimation Joseph G. Makin and Philip N. Sabes Center for Integrative Neuroscience/Department of Physiology University of California, San Francisco San Francisco, CA 94143-0444

More information

Multiple Description Coding for quincunx images.

Multiple Description Coding for quincunx images. Multiple Description Coding for quincunx images. Application to satellite transmission. Manuela Pereira, Annabelle Gouze, Marc Antonini and Michel Barlaud { pereira, gouze, am, barlaud }@i3s.unice.fr I3S

More information

On convergence of Approximate Message Passing

On convergence of Approximate Message Passing On convergence of Approximate Message Passing Francesco Caltagirone (1), Florent Krzakala (2) and Lenka Zdeborova (1) (1) Institut de Physique Théorique, CEA Saclay (2) LPS, Ecole Normale Supérieure, Paris

More information