ECE INFORMATION THEORY. Fall 2011 Prof. Thinh Nguyen

Transcription

1 1 ECE INFORMATION THEORY Fall 2011 Prof. Thinh Nguyen

2 LOGISTICS Prerequisites: ECE 353, strong mathematical background Office hours: M, W 11-Noon, KEC3115 Course Content: Entropy, Relative Entropy, and Mutual Information Asymptoptic Equipartition Property Entropy Rates of a Stochastic ti Process Data Compression Shannon Coding Theorems Channel Capacity Differential Entropy Gaussian Channel Rate Distortion Theory (time permitted, most likely not) 2

3 LOGISTICS Textbook: Elements of Information Theory, Thomas Cover and Joy Thomas, Wiley, second edition i Recommended reading: Information Theory, Inference, and Learning Algorithms, D. MacKay A Mathematical Theory of Communication, C. E. Shannon, Vol 27, pp , , July, October, 1948, labs.com/cm/ms/what/shannonday/shannon1948.pdf Class list: ece566-f11@engr.orst.edu 3

4 LOGISTICS Grading Policy Scribing: 5% Homework 30% Midterm (Take home) 30% Final exam/project (Take home) 35% 4

5 WHAT IS INFORMATION? Information theory deals with the concept of information, its measurement and its applications. 5

6 ORIGIN OF INFORMATION THEORY Two schools of thoughts British Semantic information : related to the meaning of messages Pragmatic information : related to the usage and effect of messages American Syntactic: related to the symbols from which messages are composed, and their interrelations 6

7 ORIGIN OF INFORMATION THEORY Consider the following sentences 1. Jacqueline was awarded the gold medal by the judges at the national skating competition. 2. The judges awarded Jacqueline the gold medal at the national skating competition. 3. There is a traffic jam on I-5 between Corvallis and Eugene in Oregon 4. There is a traffic jam on I-5 in Oregon (1) and (2): syntactically different but semantically and pragmatically identical. (3) and (4): syntactically and semantically different, (3) gives more precise information than (4) (3) and (4) are irrelevant for Californians. 7

8 ORIGIN OF INFORMATION THEORY The British tradition is closely related to philosophy, psychology and biology. Influenced scientists include MacKay, Carnap, Bar-Hillel, Very hard if not impossible ibl to quantify The American tradition deals with the syntactic aspects of information. Influenced scientists include Shannon, Renyi, Gallager and Csiszar, Can be rigorously quantified by mathematics 8

9 ORIGIN OF INFORMATION THEORY Basic questions of information theory in the American tradition involve the measurement of syntactic information The fundamental limits on the amount of information which can be transmitted The fundamental limits on the compression of information which can be achieved How to build information processing systems approaching these limits? 9

10 ORIGIN OF INFORMATION THEORY H. Nyquist (1924) published an article wherein he discussed how messages (characters) could be sent over a telegraph channel with maximum possible speed, but without distortion. R. Hartley (1928) who first defined a measure of information as the logarithm of the number of distinguishable messages one can represent using n characters, each can take s possible letters. 10

11 ORIGIN OF INFORMATION THEORY For a message of length n, we have the Hartley s measure of information as: H H n n ( s ) log( s ) nlog( s ) For a message of length of 1, we have the Hartley s measure of information as: H H ( s 1 ) log( s 1 ) 1log( s) This definition corresponds with the intuitive idea that a message consisting of n symbols contains n times as much information as a message consisting of only one symbol. 11

12 ORIGIN OF INFORMATION THEORY f ( s ) nf ( s ) Note that t any function would satisfy our intuition. One can show that the only functions that satisfy such equation is of the form: n f ( s) log ( s) a The choice of a is arbitrary, and is more a matter of normalization. 12

13 AND NOW, THE SHANNON S INFORMATION THEORY (a.k.a communication theory, mathematical information theory, or in short as information theory) 13

14 CLAUDE SHANNON The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. (Claude Shannon 1948) Channel Coding Theorem: It is possible to achieve near perfect communication of information over a noisy channel. In this course we will: Define what we mean by information Show how we can compress the information in a source to its theoretically minimum value and show the tradeoff between data compression and distortion. Prove the Channel Coding Theorem and derive the information capacity of different channels

15 The little formula that starts it all x A H( X) E[ log2 ( X)] p( x)log2 p( x) 15

16 SOME FUNDAMENTAL QUESTIONS (that can be addressed by information theory) What is the minimum number of bits that can be used to represent the following texts? "There is a wisdom that is woe; but there is a woe that is madness. And there is a Catskill eagle in some souls that can alike dive down into the blackest gorges, and soar out of them again and become invisible in the sunny spaces. And even if he for ever flies within the gorge, that gorge is in the mountains; so that even in his lowest swoop the mountain eagle is still higher than other birds upon the plain, even though h they soar." - Moby Dick, Herman Melville 16

17 SOME FUNDAMENTAL QUESTIONS (that can be addressed by information theory) How fast can information be sent reliably over an error prone channel? "Thera is a wisdem that is woe; but thera is a woe that is madness. And there is a Catskill eagle in sme souls that can alike divi down into the blackst goages, and soar out of thea agein and become invisble in the sunny speces. And even if he for ever flies within the gorge, that gorge is in the mountains; so that even in his lotest swoap the mountin eagle is still highhr than other berds upon the plain, even though h they soar." - Moby Dick, Herman Melville 17

18 FROM QUESTIONS TO APPLICATIONS 18

19 SOME NOT SO FUNDAMENTAL QUESTIONS (that can be addressed by information theory) How to make money? (Kelly s Portfolio Theory) What is the optimal dating strategy? 19

20 FUNDAMENTAL ASSUMPTION OF SHANNON S INFORMATION THEORY Physical Model (Laws) E.g., Newtonian universe is deterministic, all events can be predicted d with 100% accuracy by Newton s laws, given the initial conditions. U N I V E R S E If there is no irreducible randomness, the description/history of our universe, therefore can be compressed into a few equations. 20

21 FUNDAMENTAL ASSUMPTION OF SHANNON S INFORMATION THEORY Probabilistic model Some well-defined stochastic process that generates all the observations U N I V E R S E Thus in certain sense, Shannon information theory is somewhat unsatisfactory since it is based on the ignorant model. Nevertheless, it is a good approximation, depending on how accurate the probabilistic model used to describe the phenomenon of interest. 21

22 FUNDAMENTAL ASSUMPTION OF SHANNON S INFORMATION THEORY "There is a wisdom that is woe; but there is a woe that is madness. And there is a Catskill eagle in some souls that can alike dive down into the blackest gorges, and soar out of them again and become invisible in the sunny spaces. And even if he for ever flies within the gorge, that gorge is in the mountains; so that even in his lowest swoop the mountain eagle is still higher than other birds upon the plain, even though they soar." - Moby Dick, Herman Melville We know the texts above is not random, but we can build a probabilistic model for it. For example, a first order approximation could be to build a histogram of different letters in the text, t then approximate the probability that a particular letter appears based on the histogram (assuming iid). 22

23 WHY SHANNON INFORMATION IS SOMEWHAT UNSATISFACTORY? How much Shannon information does this picture contain? 23

24 ALGORITHMIC INFORMATION THEORY (KOLMOGOROV COMPLEXITY) The Kolmogorov complexity K(x) of a sequence x is the minimum size of the program and its inputs needed d to generate x. Example: If x was a sequence of all ones, a highly compressible sequence, the program would simply be a print statement t t in a loop. On the other extreme, if x were a random sequence with no structure, then the only program that could generate it would contain the sequence itself. 24

25 REVIEW OF BASIC PROBABILITY A discrete random variable X takes a value x from the alphabet A with probability p(x). 25

26 EXPECTED VALUE If g(x) is real valued and defined on A then X E [ g( X)] p( x) g( x) x A 26

27 SHANNON INFORMATION CONTENT The Shannon Information Content of an outcome with probability p is log 2 p 27

28 MINESWEEPER Where is the bomb? 16 possibilities - needs 4 bits to specify 28

29 ENTROPY H ( X ) E [ log 2 ( p ( X ))] p ( x)log 2 ( p ( x )) 2 x A 29

30 ENTROPY EXAMPLES Bernoulli Random Variable 30

31 ENTROPY EXAMPLES Four Colored Shapes A = [ ; ; ; ] p(x) = [1/2;1/4;1/8;1/8] 31

32 DERIVATION OF SHANNON ENTROPY H ( X ) n i 1 p i log 2 p i Intuitive Requirements: 1. We want H to be a continuous function of probabilities p i. That is, a i small change in p i should only cause a small change in H 2. If all events are equally likely, that is, p i = 1/n for all i, then H should be a monotonically increasing function of n. The more possible outcomes there are, the more information should be contained in the occurrence of any particular outcome. 3. It does not matter how we divide the group of outcomes, the total of information should be the same. 32

33 DERIVATION OF SHANNON ENTROPY 33

34 DERIVATION OF SHANNON ENTROPY 34

35 DERIVATION OF SHANNON ENTROPY 35

36 DERIVATION OF SHANNON ENTROPY 36