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Unformatted text preview: 1 Chapter 14 Entropy Teacher: è ! ª Office: 805 Tel: ext. 31822 Email: [email protected] 2 141 Introduction Probability of an event A , denoted as ( ) P A , w event AP P uncertainty1 measure1 If ( ) 1, no unceretainty at all If ( ) 0, P A A P A A = ⇒ = h ( ) 0.5 P A = A  * w 3 A P uncertainty1 maximum 3 EntropyH w 3 experiment S 1 partition U 1 uncertainty measure Probability space S P elementary events (outcomes) 1 Partition { } 1 , , N U A A = 1 2 N A A A S ∪ ∪ ∪ = i j A A φ ∩ = Entropy of U is defined as 1 1 ( ) log log N N H U p p p p =    , where ( ) i i p P A = Information = uncertainty1  3w 3 H w 3 information  3w 3 information 4 5 Ex. 41 1 fairdie 1 experiment 1 (a) 1 { , } U even odd = { } { } 0.5 P even P odd = = Hence 1 1 1 1 ( ) log log log 2 2 2 2 2 H U =  = (b) 1 V P elementary events { } i V f = 8 w 3 1 { } 6 i P f = ( ) log 6 H V = Entropy 1 log31 uncertainty about V assuming U (conditional entropy)1  3 even 1 oddk w uncertainty 1 log3. Ex. 142 For coin experiment1 { } P h p = ( ) log (1 )log(1 ) ( ) H V p p p p h p =  ≡ Fig.142 1 ( ) h p verse pP p =0.5 1 maximum1 p= 0 1 1 1 01 6 7 142 Basic Concepts Let { } { } 1 , , N i U A A A = = 1 partition of SP i AP events (1) w events1 binary , In this case, we usually denote U by { } , U A A = A is called the complement of A (2) 1 UP events 1 elementary events { } i ξ denote it by V and call it element partition 8 (3) A refinement of U is a partition B of S , such that for j B B 2200 ∈ j B P i A P subset1 denoted by B U P , 1 B P U P events: events (see Fig.144)1 h B U P iff j i B A ⊂ A common refinement 1 partitions 1 refinement (4) The product of U and B is a partition1 elements 1 i j AB (1 )1 denoted by U B ⋅ U B ⋅ U P B P common refinement 9 10 11 Properties V U P for any U U B B U ⋅ = ⋅ , ( ) ( ) U B C U B C ⋅ ⋅ = ⋅ ⋅ If 1 2 3 U U U P , then 1 3 U U P If B U P , then U B B ⋅ = 12 Entropy Definition: 1 1 1 ( ) ( log log ) ( ) N N N i i H U p p p p p ϕ = =  + + = ∑ where ( ) i i p P A = ( ) log p p p ϕ =  h ( ) p ϕ ≥ for 0 1 p ≤ ≤ , ∴ ( ) H U ≥ ( ) H U = if 1 1 i p = i p = 13 For binary partition1 ( ) p P A = ( ) 1 P A p =  ∴ ( ) log (1 )log(1 ) ( ) H U p p p p h p =     = If 1 1 2 N p p p p = = = = then 1 1 1 1 ( ) log log log H U N N N N N =    = In this case, if 2 m N = , then ( ) H U m = . (Here, the base of log operation is 2) Inequality1 ( ) log p p p ϕ =  convex function 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) p p p p p p ϕ ϕ ϕ ϕ ε ϕ ε + < + < + +  where 1 1 2 2 p p p p ε ε < + ≤  < . !≡♠ 3 ( ) p ϕ (see Fig.146) 14 15 h inequalityw 3 (1) If { } 1 2 , , , N U A A A = and { } 2 , , , , a b N B B B A A = where a B 1 b B 1 1 A split 1 1 1 events1 (see Fig.147) 1 ( ) ( ) H U H B ≤ , 1 1 2 1 1 2 ( ) ( ) ( ), ( ( )) ( ( )) ( ( )) P A P B P B P A P B P B ϕ ϕ ϕ = + ≤ + (convex function 1 ) 16...
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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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