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Unformatted text preview: Chapter14 Entropy 141 Introduction The probability of an event A , denoted as ( ) P A , can be interpreted as an uncertainty measure of the occurrence of the event. If ( ) 1, we are definitely sure that will occur no unceretainty at all If ( ) 0, we are definitely sure that will not occur P A A P A A = = On the other hand, for ( ) 0.5 P A = , the probabilities that A will occur and that A will not occur are equal. We therefore can say that the uncertainty of the event A is maximum. Entropy is defined as an uncertainty measure of a partition U of the probability space of an experiment S . The probability space S is the set of all elementary events (or outcomes). For a partition { } 1 , , N U A A = L , 1 2 N A A A S = L and i j A A φ = . Entropy of U is defined as 1 1 ( ) log log N N H U p p p p =  L , where ( ) i i p P A = . Information = uncertaintyh It is a viewpoint that the occurrence of an uncertain event brings more information while a certain event (either occur or not occur) carries no information. Ex. 41: Experiment of tossing fairdie: (a) Let { , } U even odd = . Obviously, { } { } 0.5 P even P odd = = . Hence 1 1 1 1 ( ) log log log2 2 2 2 2 H U =  = (b) Let V be the set of elementary events { } i V f = . Obviously, 1 { } 6 i P f = . Then, ( ) log6 H V = . The difference of their entropy is log3. It is the uncertainty about V assuming that U is known. This means that if we know that the outcome is even (or odd), then the uncertainty of outcome reduces to log3. It is known as conditional entropy . Ex. 142: For a coin experiment, { } P h p = . Then ( ) log (1 )log(1 ) ( ) H V p p p p h p =  C Fig.142 displays ( ) h p verse p . The maximum occurs at p =0.5. It is 0 at p= 0 and p =1. 142 Basic Concepts Let { } { } 1 , , N i U A A A = = L be a partition of S , where , 1, , , i A i N = L are events. (1) If there are only two events, we call it a binary case. Usually, U is denoted as { } , U A A = , were A is called the complement of A . (2) If the events of U are all elementary events { } i ξ , we denote it by V and call it element partition. (3) A refinement of U is a partition B of S , such that for j B B 2200 , j B is a subset of some event i A . It is denoted by B U p . In other words, some events i A are divided into subevents contained in B . (4) The product of two partitions U and B is a partition which contains all intersection i j A B of their elements, and is denoted by U B P . U B P is the largest common refinement of U and B . Properties V U p is true for any U . U B B U = If 1 2 3 U U U p p , then 1 3 U U p . If B U p , then U B B = . Entropy Definition: 1 1 1 ( ) ( log log ) ( ) N N N i i H U p p p p p ϕ = =  + + = L where ( ) i i p P A = and ( ) log p p p ϕ =  ....
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This note was uploaded on 07/21/2009 for the course CM EM5102 taught by Professor Sinhorngchen during the Fall '08 term at National Chiao Tung University.
 Fall '08
 SinHorngChen

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