ELEMENT OF INFORMATION THEORY

Transcription

1 History Table of Content ELEMENT OF INFORMATION THEORY O. Le Meur Univ. of Rennes 1 October

2 History Table of Content VERSION: : Document creation, done by OLM; : Document updated, done by OLM: Slide in introduction: The human communication; Slide on Spearman's rank correlation coecient; Slide on Correlation does not imply causation; Slide to introduce the amount of information; Slide presenting a demonstration concerning the joint entropy H(X, Y ). 2

3 History Table of Content ELEMENTS OF INFORMATION THEORY

4 Information Theory Goal and framework of the communication system Some denitions 1 Goal and framework of the communication system Some denitions

5 The human communication... Goal and framework of the communication system Some denitions Visual signal: the smoke signal is one of the oldest forms of communication in recorded history; Painting by Frederic Remington. Smoke signals were also employed by Hannibal, the Carthaginian general who lived two centuries BC Sonor signal: in Ancient China, soldiers stationed along the Great Wall would alert each other of impending enemy attack by signaling from tower to tower. In this way, they were able to transmit a message as far away as 480 km (300 miles) in just a few hours. A common point, there is a code... 5

6 Goal and Framework of the communication system Goal and framework of the communication system Some denitions To transmit an information at the minimum rate for a given quality; Seminal work of Claude Shannon (1948)[Shannon,48]. Ultimate goal The source and channel codec must be designed to ensure a good transmission of a message given a minimum bit rate or a minimum level of quality. 6

7 Goal and framework of the communication system Goal and framework of the communication system Some denitions Three major research axis: 1 Measure: carried by a message. 2 Compression: Lossy vs lossless coding... Mastering the distortion d(s, Ŝ) 3 Transmission: Channel and noise modelling Channel capacity 7

8 Some denitions Goal and framework of the communication system Some denitions Denition Source of information: something that produces messages! Denition Message: a stream of symbols taking theirs values in a predened alphabet. Example Source: a camera Message: a picture Symbols: RGB coecients alphabet= (0,..., 255) Example Source: book Message: a text Symbols: letters alphabet= (a,..., z) 8

9 Some denitions Goal and framework of the communication system Some denitions Denition Source Encoder: the goal is to transform S in a binary signal X of size as small as possible (eliminate the redundancy). Channel Encoding: the goal is to add some redundancy in order to be sure to transmit the binary signal X without errors. Denition Compression Rate: σ = Nb bits of input Nb bits of output 9

10 Information Theory Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient 1 2 Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient

11 Random variables and probability distribution Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient The transmitted messages are considered as a random variable with a nite alphabet. Denition (Alphabet) An alphabet A is a set of data {a 1,..., a N } that we might wish to encode. Denition (Random Variable) A discrete random variable X is dened by an alphabet A = {x 1,..., x N } and a probability distribution {p 1,..., p N }, i.e. p i = P(X = x i ). Properties Remark: a symbol is the outcome of a random variable. 0 p i 1; N i=1 p i = 1 also noted x A p(x). p i = P(X = x i ) is equivalent to P X (x i ) and P X (i). 11

12 Joint probability Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Denition (Joint probability) Let X and Y be discrete random variables dened by alphabets {x 1,..., x N } and {y 1,..., y M }, respectively. A and B are the events X = x i and Y = y j, P(X = x i, Y = y j ) is the joint probability also called P(A, B) or p ij. Properties of the joint probability density function (pdf) N M P(X = x i=1 j=1 i, Y = y j ) = 1, If A B =, P(A, B) = P(X = x i, Y = y j ) = 0, Marginal probability distribution of X and Y : P(A) = P(X = x i ) = M j=1 P(X = x i, Y = y j ) is the probability of the event A, P(B) = P(Y = y j ) = N i=1 P(X = x i, Y = y j ) is the probability of the event B. 12

13 Joint probability Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Example Let X and Y be discrete random variables dened by alphabets {x 1, x 2} and {y 1, y 2, y 3, y 4}, respectively. The sets of events of (X, Y ) can be represented in a joint probability matrix: X, Y y 1 y 2 y 3 y 4 x 1 (x 1, y 1) (x 1, y 2) (x 1, y 3) (x 1, y 4) x 2 (x 2, y 1) (x 2, y 2) (x 2, y 3) (x 2, y 4) 13

14 Joint probability Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Example Let X and Y be discrete random variables dened by alphabets {R, NR} and {S, Su, A, W }, respectively. X, Y S Su A W R NR

15 Joint probability Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Example Let X and Y be discrete random variables dened by alphabets {R, NR} and {S, Su, A, W }, respectively. Questions: X, Y S Su A W R NR Does (X, Y ) dene a pdf? => Yes, 2 i=1 4 j=1 P(X = x i, Y = y j ) = 1; Is it possible to dene the marginal pdf of X? => Yes, X = {R, NR}, P(X = R) = 4 j=1 P(X = R, Y = y j) = 0.55, P(X = NR) = 4 j=1 P(X = NR, Y = y j) =

16 Conditional probability and Bayes rule Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Notation: the conditional probability of X = x i knowing that Y = y j is written as P(X = x i Y = y j ). Denition (Bayes rule) P(X = x i Y = y j ) = P(X =x i,y =y j ) P(Y =y j ) Properties N k=1 P(X = x k Y = y j ) = 1; P(X = x i Y = y j ) P(Y = y j X = x i ). 14

17 Conditional probability and Bayes rule Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Example Let X and Y be discrete random variables dened by alphabets {R, NR} and {S, Su, A, W }, respectively. X, Y S Su A W R NR Question: What is the conditional probability distribution of P(X = x i Y = S)? 15

18 Conditional probability and Bayes rule Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Example Let X and Y be discrete random variables dened by alphabets {R, NR} and {S, Su, A, W }, respectively. X, Y S Su A W R NR Question: What is the conditional probability distribution of P(X = x i Y = S)? P(Y = S) = 2 P(Y = y i=1 i ) = 0.25, and from Bayes P(X = R Y = S) = and P(X = NR Y = S) =

19 Statistical independence of two random variables Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Denition (Independence) Two discrete random variables are independent if the joint pdf is egal to the product of the marginal pdfs: P(X = x i, Y = y j ) = P(X = x i ) P(Y = y j ) i and j. Remark: If X and Y independent, P(X = x i Y = y j ) = P(X = x i ) (From Bayes). While independence of a set of random variables implies independence of any subset, the converse is not true. In particular, random variables can be pairwise independent but not independent as a set. 16

20 Linear correlation coecient (Pearson) Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Denition (linear correlation coecient) The degree of linearity between two discrete random variables X and Y is given by the linear correlation coecient r: r(x, Y ) def = COV (X,Y ) σ X σ Y COV (X, Y ) is the covariance given by 1 N (X X )(Y Y ) (X σ X is the standard deviation given by X ). N Properties r(x, Y ) [ 1, 1]; r(x, Y ) 1 means that X and Y are very correlated; r(t X, Y ) = r(x, Y ), where T is a linear transform. 17

21 Linear correlation coecient (Pearson) Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Relationship between the regression line and the linear correlation coecient: Regression line The goal is to nd Ŷ = ax + b. a and b deduced from the minimization of the MSE: J = argmin (a,b) (Y Ŷ ) 2 J COV (X,Y ) = 0 a = a σ X 2 J b = 0 b = Y ax The slope a is equal to r, only when σ X = σ Y (standardised variables). 18

22 Spearman's rank correlation coecient Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Denition (Spearman's rank correlation coecient) The degree of linearity between two discrete random variables X and Y is given by the linear correlation coecient r: ρ(x, Y ) def = 1 6 d 2 i n 2 (n 1) where d i = x i y i. The n data X i and Y i are converted to ranks x i and y i. To illustrate the dierence between Pearson and Spearman correlation coecient: Pearson correlation is 0.456; Spearman correlation is 1. (0, 1),(10, 100),(101, 500),(102, 2000) 19

23 Correlation does not imply causation Random variables and probability distribution Joint probability Conditional probability and Bayes rule Statistical independence of two random variables Correlation coecient Correlation does not imply causation is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other. Example (Ice cream and drowning) As ice cream sales increase, the rate of drowning deaths increases sharply. Therefore, ice cream causes drowning. The aforementioned example fails to recognize the importance of time in relationship to ice cream sales. Ice cream is sold during the summer months at a much greater rate, and it is during the summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water based activities, not ice cream. From 20

24 Information Theory Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram

25 Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Claude Shannon ( ) and communication theory For Shannon a message is very informative if the chance of its occurrence is small. If, in contrast, a message is very predictable, then it has a small amount of information. One is not surprised to receive it. A measure of the amount of information in a message. 22

26 Self-Information Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Let X be a discrete random variable dened by the alphabet {x 1,..., x N } and the probability density {p(x = x 1),..., p(x = x N )}. How to measure the amount of information provided by an event A, X = x i? Denition (Self-Information proposed by Shannon) I (A) def = log 2p(A) I (X = x i ) def = log 2p(X = x i ) Unit: bit/symbol. Properties I (A) 0; I (A) = 0 if p(a) = 1; if p(a) < p(b) then I (A) > I (B); p(a) 0, I (A) +. The self-information can be measured in bits, nits, or hartleys depending on the base of the logarithm. 23

27 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Shannon introduced the entropy rate, a quantity that measured a source's information production rate. Denition (Shannon Entropy) The entropy of a discrete random variable X dened by the alphabet {x 1,..., x N } and the probability density {p(x = x 1),..., p(x = x N )} is given by: H(X ) = E{I (X )} = N i=1 p(x = x i )log 2(p(X = x i )), unit: bit/symbol. The entropy H(X ) is a measure of the amount of uncertainty, a measure of surprise associated with the value of X. Properties Entropy gives the average number of bits per symbol to represent X H(X ) 0; H(X ) log 2N (equality for a uniform probability distribution). 24

28 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example Example 1: The value of p(0) is highly predictable, the entropy (amount of uncertainty) is zero. p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) Entropy (bits/symbol) Example

29 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example Example 1: The value of p(0) is highly predictable, the entropy (amount of uncertainty) is zero; Example 2: This is a probability distribution of Bernoulli {p, 1 p}. p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) Entropy (bits/symbol) Example Example

30 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example Example 1: The value of p(0) is highly predictable, the entropy (amount of uncertainty) is zero; Example 2: This is a probability distribution of Bernoulli {p, 1 p}; Example 3: Uniform probability distribution (P(X = x i ) = M 1, with M = 4, i {2,..., 5}). p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) Entropy (bits/symbol) Example Example Example

31 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example Example 1: The value of p(0) is highly predictable, the entropy (amount of uncertainty) is zero; Example 2: This is a probability distribution of Bernoulli {p, 1 p}; Example 3: Uniform probability distribution (P(X = x i ) = M 1, with M = 4, i {2,..., 5}); Example 4: ; p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) Entropy (bits/symbol) Example Example Example Example

32 Shannon entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example Example 1: The value of p(0) is highly predictable, the entropy (amount of uncertainty) is zero; Example 2: This is a probability distribution of Bernoulli {p, 1 p}; Example 3: Uniform probability distribution (P(X = x i ) = M 1, with M = 4, i {2,..., 5}); Example 4: ; Example 5: Uniform probability distribution (P(X = x i ) = N 1, with N = 8, i {0,..., 7}) p(0) p(1) p(2) p(3) p(4) p(5) p(6) p(7) Entropy (bits/symbol) Example Example Example Example Example

33 Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example (Shannon entropy) (a) H=1.39bit/s (b) H=3.35bit/s (c) H=6.92bit/s (d) H=7.48bit/s (e) H=7.62bit/s (c) Original picture (256 levels); (a) Original picture (4 levels); (b) Original picture (16 levels); (d) Original picture + uniform noise; (e) Original picture + uniform noise (more than previous one). 26

34 Joint information/joint entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Denition (Joint information) Let X and Y be discrete random variables dened by alphabet {x 1,..., x N } and {y 1,..., y M }, respectively. The joint information of two events (X = x i ) and (Y = y i ) is dened by I (X = x i, Y = y j ) = log 2(p(X = x i, Y = y j )). Denition (Joint entropy) The joint entropy of the two discret random variables X and Y is given by: H(X, Y ) = E{I (X = x i, Y = y j )}. H(X, Y ) = N i=1 M j=1 p(x = x i, Y = y j )log 2(p(X = x i, Y = y j )) 27

35 Joint information/joint entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Demonstrate that the joint entropy H(X, Y ) is equal to H(X ) + H(Y ), if X and Y are statistically independant: H(X, Y ) = P(x, y) log P(x, y) x X y Y = P(x)P(y) log P(x)P(y) x X y Y = P(x)P(y) [log P(x) + log P(y)] x X y Y = P(x)P(y) log P(x) P(x)P(y) log P(y) x X y Y y Y x X = P(x) log P(x) P(y) P(y) log P(y) P(x) x X y Y y Y x X = P(x) log P(x) P(y) log P(y) x X y Y H(X, Y ) = H(X ) + H(Y ) 28

36 Joint information/joint entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram In the same vein, you should know how to demonstrate that: H(X, Y ) H(X ) + H(Y ) (Equality if X and Y independant). This property is called sub-additivity. It means that two systems, considered together, can never have more entropy than the sum of the entropy in each of them. 29

37 Conditional information/conditional entropy Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Denition (Conditional information) The conditional information... I (X = x i Y = y j ) = log 2(p(X = x i Y = y j )) Denition (Conditional entropy) The conditional entropy of Y given the random variable X : H(Y X ) = H(Y X ) = N p(x = x i )H(Y X = x i ) i=1 N M i=1 j=1 p(x = x i, Y = y j )log 2 1 p(y = y j X = x i ) The conditional entropy H(Y X ) is the amount of uncertainty remaining about Y after X is known. Remarks: We always have H(Y X ) H(Y ) (The knowledge reduces the uncertainty); Entropy chain rule: H(X, Y ) = H(X ) + H(Y X ) = H(Y ) + H(X Y ) (From Bayes); 30

38 Mutual information Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Denition (Mutual information) The Mutual information of two events X = x i and Y = y j is dened as I (X = x i ; Y = y j ) = log 2 p(x =x i )p(y =y j ) p(x =x i,y =y j ) The Mutual information of two random variables X and Y is dened as I (X ; Y ) = N i=1 Mj=1 p(x = x i, Y = y j )log 2 p(x =x i )p(y =y j ) p(x =x i,y =y j ) The mutual information I (X ; Y ) measures the information that X and Y share... Be careful to the notation, I (X, Y ) I (X ; Y ) Properties Symmetry: I (X = x i ; Y = y j ) = I (Y = y j ; X = x i ); I (X ; Y ) 0; zero if and only if X and Y are independent variables; H(X X ) = 0 H(X ) = I (X ; X ) I (X ; X ) I (X ; Y ). 31

39 Mutual information Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Mutual information and dependency The mutual information can be expressed as : I (X ; Y ) = D(p(X = x i, Y = y j ) p(x = x i )p(y = y j )) where, p(x = x i, Y = y j ) joint pdf of X and Y ; p(x = x i ) and p(y = y i ) marginal probability distribution of X and Y, respectively; D(..) the divergence of Kullback-Leibler. Remarks regarding the KL-divergence: D(p q) = N i=1 p(x = x i )log 2 p(x =x i ) p(q=q i ), Q random variable {q 1,..., q N }; This is a measure of divergence between two pdfs, not a distance; D(p q) = 0, if and only if the two pdfs are strictly the same. 32

40 Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram Example (Mutual information) Mutual information between the original picture of C. Shannon and the following ones: (a) I=2.84 (b) I=2.54 (c) I=2.59 (d) I=2.41 (e) I=1.97 (a) Original picture + uniform noise; (b) Original picture + uniform noise (more than previous one); (c) Human; (d) Einstein; (e) A landscape. 33

41 Venn's diagram Self-Information Entropy denition Joint information, joint entropy Conditional information, conditional entropy Mutual information Venn's diagram X Y H(X Y ) I (X ; Y ) H(Y X ) We retrieve: H(X, Y ) = H(X ) + H(Y X ) H(X, Y ) = H(Y ) + H(X Y ) H(X, Y ) = H(X Y ) + H(Y X ) + I (X ; Y ) I (X ; Y ) = H(X ) H(X Y ) I (X ; Y ) = H(Y ) H(Y X ) H(X ) H(Y ) H(X, Y ) 34

42 Information Theory Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source)

43 Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Remind of the goal To transmit an information at the minimum rate for a given quality; Seminal work of Claude Shannon (1948)[Shannon,48]. 36

44 Parameters of a discrete source Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Denition (Alphabet) An alphabet A is a set of data {a 1,..., a N } that we might wish to encode. Denition (discrete source) A source is dened as a discrete random variable S dened by the alphabet {s 1,..., s N } and the probability density {p(s = s 1),..., p(s = s N )}. Example (Text) Alphabet={a,..., z} Source Message {a, h, y, r, u} 37

45 Discrete memoryless source Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Denition (Discrete memoryless source) A discrete source S is memoryless if the symbols of the source alphabet are independent and identically distributed: p(s = s 1,..., S = s N ) = N i=1 p(s = s i ) Remarks: Entropie: H(S) = N i=1 p(s = s i )log 2p(S = s i ) bit; Particular case of a uniform source: H(S) = log 2N. 38

46 Extension of a discrete memoryless source Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Rather than considering individuals symbols, more useful to deal with blocks of symbols. Let S be a discrete source with an alphabet of size N. The output of the source is grouped into blocks of K symbols. The new source, called S K, is dened by an alphabet of size N K. Denition (Discrete memoryless source, K th extension of a source S) If the source S K is the K th extension of a source S, the entropy per extended symbols of S K is K times the entropy per individual symbol of S: H(S K ) = K H(S) Remark: the probability of a symbol si K p(si K ) = K j=1 p(s ij ). = (s i1,..., s ik ) from the source S K is given by 39

47 with memory (Markov source) Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Discrete memoryless source This is not realistic! Successive symbols are not completely independent of one another... in a picture: a pel (S 0) depends statistically on the previous pels s5 s4 s3 s2 s1 s0 This dependence is expressed by the conditionnal probability p(s 0 S 1, S 2, S 3, S 4, S 5). p(s 0 S 1, S 2, S 3, S 4, S 5) p(s 0) in the langage (french): p(s k = u) p(s k = e), p(s k = u S k 1 = q) >> p(s k = e S k 1 = q); 40

48 with memory (Markov source) Parameters of a discrete source Discrete memoryless source Extension of a discrete memoryless source with memory (Markov source) Denition ( with memory) A discrete source with memory of order N (N th order Markov) is dened as: p(s k S k 1, S k 2,..., S k N ) The entropy is given by: H(S) = H(S k S k 1, S k 2,..., S k N ) Example (One dimensional Markov model) The pel value S 0 depends statistically only on the pel value S 1. Q H(X ) = 1.9 bit/symb, H(Y ) = 1.29, H(X, Y ) = 2.15, H(X Y ) =

49 Information Theory Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy 6 42

50 Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Remind of the goal To transmit an information at the minimum rate for a given quality; Seminal work of Claude Shannon (1948)[Shannon,48]. 43

51 Parameters of a discrete channel Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy X t {x 1,..., x N } Communication channel Y t {y 1,..., y M } Denition (Channel) A channel transmits as best as it can symbols X t. Symbols Y t are produced (possibly corrupted). The subscript t indicates the time. The channel is featured by the following conditional probabilities: p(y t = y j X t = x i, X t = x i 1...) The output symbol can be dependent on several input symbols. Capacity The channel is characterized by its capacity C bits per symbol. It is an upper bound on the bit rate that can be accomodated by the channel. C = max p(x ) I (X ; Y ) 44

52 Memoryless channel Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Denition (Memoryless channel) The channel is memoryless when the conditionnal probability p(y X ) is independent of previously transmitted channel symbols: p(y = y j X = x i ). Notice that the subscript t is no more required... Transition matrix Π, size (N, M): p(y =y1 X =x1)... p(y =y m X =x1) Π(X, Y ) = p(y =y1 X =x n)... p(y =y m X =x n) Graphical representation: x1 p(y1 x 1 ) p(y2 x 1 ) y1 x2 y2 p(ym x1 ) xn p(ym xn ) ym 45

53 Ideal Channel Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Ideal Channel or noiseless channel Perfect correspondance between the input and the output. Each transmitted bit is received without error. p(y = y j X = x i ) = δ xi y j with, δ xy = 1 if x = y 0 if x y The channel capacity is C = max p(x ) I (X ; Y ) = 1bit. Π(X, Y ) = ( )

54 Binary Symmetric Channel (BSC) Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Denition (Binary Symmetric Channel (BSC)) A binary symmetric channel has binary input and binary output (X = {0, 1}, Y = {0, 1}). C = 1 H(p) bits. p the probability that the symbol is modied at the ouptut. ( ) 1 p p Π(X, Y ) = p 1 p p p p 1 p 0 1 From left to right: original binary signal, result for p = 0.1, p = 0.2 and p = 1 (the 2 channel has a null capacity!). 47

55 Binary Erasure Channel (BEC) Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Denition (Binary Erasure Channel (BEC)) A binary erasure channel has binary input and ternary output (X = {0, 1}, Y = {0, 1, e}). The third symbol is called an erasure (complete loss of information about an input bit). C = 1 p e p bits. p e is the probability to receive the symbol e. 1 p p e 0 0 p e ( 1 p pe p e p Π(X, Y ) = p p e 1 p p e ) p p p e e 1 1 p p e 1 48

56 Channel with memory Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Gilbert-Elliott model The channel is a binary symmetric channel with memory determined by a two states Markov chain. A channel has two states: G, good state and B, bad state. α and β are the transition probabilities α 1 β G B 1 α β Transition matrix to go from one state to another: S t = {G, B}. S t ( 1 β S t 1 β ) α 1 α 49

57 : a taxonomy Parameters of a discrete channel Memoryless channel Channel with memory A taxonomy Lossless channel: H(X Y ) = 0, the input is dened by the output; Deterministic channel: H(Y X ), the output dened the input; Distortionless channel: H(X ) = H(Y ) = I (X ; Y ), deterministic and lossless; Channel with a null capacity: I (X ; Y ) = 0. 50

58 Information Theory Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem 51 7

59 Source code Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Denition (Source code) A source code C for a random variable X is a mapping from x X to {0, 1}. Let c i denotes the code word for x i and l i denote the length of c i. {0, 1} is the set of all nite binary string. Denition (Prex code) A code is called a prex code (instantaneous code) if no code word is a prex of another code word Not required to wait for the whole message to be able to decode it. 52

60 Kraft inequality Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Denition (Kraft inequality) A code C is instantaneous if it satises the following inequality: N i=1 2 l i 1 with, l i the length of code word length i 53

61 Kraft inequality Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Example (Illustration of the Kraft inequality using a coding tree) The following tree contains all three-bit codes:

62 Kraft inequality Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Example (Illustration of the Kraft inequality using a coding tree) The following tree contains a prex code. We decide to use the code word 0 and The remaining leaves constitute a prex code: 4 i=1 2 l i = = 1 55

63 Higher bound of entropy Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Let S a discret source dened by the alphabet {s 1,..., s N } and the probability density {p(s = s 1),..., p(s = s N )}. Denition (Higher bound of entropy) H(S) log 2N Interpretation the entropy is limited by the size of the alphabet; a source with a uniform pdf provides the highest entropy. 56

64 Source coding theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Let S a discrete source dened by the alphabet {s 1,..., s N } and the probability density {p(s = s 1),..., p(s = s N )}. Each symbol s i is coded with a length l i bits: Denition (Source coding theorem or First ) H(S) l C with l C = N i=1 p i l i The entropy of the source gives the limit of the lossless compression. We can not encode the source with less than H(S) bit per symbol. The entropy of the source is the lower-bound. Warning... {l i } i=1,...,n must satisfy Kraft's inequality. Remarks: l C = H(S), when l i = log 2p(X = x i ). 57

65 Source coding theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Denition (Source coding theorem (bis)) Whatever the source S, there exist an instantaneous code C, such that H(S) l C < H(S) + 1 The upper bound is equal to H(S) + 1, simply because the Shannon information gives a fractionnal value. 58

66 Source coding theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Example Let X a random variable with the following probability density. The optimal code lengths are given by the self-information: X x 1 x 2 x 3 x 4 x 5 P(X = x i ) I (X = x i ) The entropy H(X ) is equal to bits. The source coding theorem gives: l <

67 Rabbani-Jones extension Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Symbols can be coded in blocks of source samples instead of one at a time (block coding). In this case, further bit-rate reductions are possible. Denition (Rabbani-Jones extension) Let S be an ergodic source with an entropy H(S). Consider encoding blocks of N source symbols at a time into binary codewords. For any δ > 0, it is possible to construct a code that the average number of bits per original source symbol l C satises: H(S) l C < H(S) + δ Remarks: Any source can be losslessly encoded with a code very close to the source entropy in bits; There is a high benet to increase the value N; But, the number of symbols in the alphabet becomes very high. Example: block of 2 2 pixels (coded on 8 bits) leads to values per block... 60

68 Channel coding theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Let a discrete memoryless channel of capacity C. The channel coding transform the messages {b 1,..., b k } into binary codes having a length n. Denition (Transmission rate) The transmission rate R is given by: R def = k n R is the amount of information stemming from the symbols b i per transmitted bits. B i block of length k Channel coding C i block of length n 61

69 Channel coding theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Example (repetition coder) The coder is simply a device repeating r times a particular bit. Below, example for r = 2. R = 1 3. B t 011 Channel coding C t This is our basic scheme to communicate with others! We repeat the information... Denition (Channel coding theorem or second ) R < C, p e > 0: it is possible to transmit information nearly without error at any rate below the channel capacity; if R C, all codes will have a probability of error greater than a certain positive minimal level, and this level increases with the rate. 62

70 Source/Channel theorem Source code Kraft inequality Higher bound of entropy Source coding theorem Rabbani-Jones extension Channel coding theorem Source/Channel theorem Let a noisy channel having a capacity C and a source S having an entropy H. Denition (Source/Channel theorem) if H < C it is possible to transmit information nearly without error. Shannon showed that it was possible to do that by making a source coding followed by a channel coding; if H C, the transmission cannot be done with an arbitrarily small probability. 63

71 Information Theory

72 YOU MUST KNOW Let X a random variable dened by X = {x 1,..., x N } and the probabilities {p x1,..., p xn }. Let Y a random variable dened by Y = {y 1,..., y N } and the probabilities {p y1,..., p yn }. Ni=1 p xi = 1 Independence: p(x = x 1,..., X = x N ) = N i=1 p(x = x i ) Bayes rule: p(x = x i Y = y j ) = p(x =x i,y =y j ) p(y =y j ) Self information: I (X = x i ) = log 2 p(x = x i ) Mutual information: I (X ; Y ) = N i=1 Mj=1 p(x = x i, Y = y j )log 2 p(x =x i )p(y =y j ) p(x =x i,y =y j ) Entropy: H(X ) = N i=1 p(x = x i )log 2 p(x = x i ) Conditional entropy of Y given X : H(Y X ) = N i=1 p(x = x i )H(Y X = x i ) Higher Bound of entropy: H(X ) log 2 N Limit of the lossless compression : H(X ) l C, l C = N i=1 p xi l i 65

73 Suggestion for further reading... [Shannon,48] C.E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal, 27,