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Unformatted text preview: Chapter 29 Entropy Rates and Asymptotic Equipartition Section 29.1 introduces the entropy rate — the asymptotic en tropy per timestep of a stochastic process — and shows that it is welldefined; and similarly for information, divergence, etc. rates. Section 29.2 proves the ShannonMacMillanBreiman theorem, a.k.a. the asymptotic equipartition property, a.k.a. the entropy ergodic theorem: asymptotically, almost all sample paths of a sta tionary ergodic process have the same logprobability per timestep, namely the entropy rate. This leads to the idea of “typical” se quences, in Section 29.2.1. Section 29.3 discusses some aspects of asymptotic likelihood, us ing the asymptotic equipartition property, and allied results for the divergence rate. 29.1 InformationTheoretic Rates Definition 376 (Entropy Rate) The entropy rate of a random sequence X is h ( X ) ≡ lim n H ρ [ X n 1 ] n (29.1) when the limit exists. Definition 377 (Limiting Conditional Entropy) The limiting conditional entropy of a random sequence X is h ( X ) ≡ lim n H ρ [ X n  X n 1 1 ] (29.2) when the limit exists. 197 CHAPTER 29. RATES AND EQUIPARTITION 198 Lemma 378 For a stationary sequence, H ρ [ X n  X n 1 1 ] is nonincreasing in n . Moreover, its limit exists if X takes values in a discrete space. Proof: Because “conditioning reduces entropy”, H ρ [ X n +1  X n 1 ] ≤ H [ X n +1  X n 2 ]. By stationarity, H ρ [ X n +1  X n 2 ] = H ρ [ X n  X n 1 1 ]. If X takes discrete values, then conditional entropy is nonnegative, and a nonincreasing sequence of non negative real numbers always has a limit. Remark: Discrete values are a sufficient condition for the existence of the limit, not a necessary one. We now need a naturallooking, but slightly technical, result from real anal ysis. Theorem 379 (Ces` aro) For any sequence of real numbers a n → a , the se quence b n = n 1 ∑ n i =1 a n also converges to a . Proof: For every > 0, there is an N ( ) such that  a n a  < whenever n > N ( ). Now take b n and break it up into two parts, one summing the terms below N ( ), and the other the terms above. lim n  b n a  = lim n n 1 n i =1 a i a (29.3) ≤ lim n n 1 n i =1  a i a  (29.4) ≤ lim n n 1 N ( ) i =1  a i a  + ( n N ( )) (29.5) ≤ lim n n 1 N ( ) i =1  a i a  + n (29.6) = + lim n n 1 N ( ) i =1  a i a  (29.7) = (29.8) Since was arbitrary, lim b n = a . Theorem 380 (Entropy Rate) For a stationary sequence, if the limiting con ditional entropy exists, then it is equal to the entropy rate, h ( X ) = h ( X ) . Proof: Start with the chain rule to break the joint entropy into a sum of conditional entropies, use Lemma 378 to identify their limit as h ] prime ( X ), and CHAPTER 29. RATES AND EQUIPARTITION 199 then use Ces` aro’s theorem: h ( X ) = lim n 1 n H ρ [ X n 1 ] (29.9) = lim n 1 n n i =1 H ρ [ X i  X i 1 1 ] (29.10) = h ( X ) (29.11) as required....
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This note was uploaded on 12/20/2011 for the course STAT 36754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
 Spring '06
 Schalizi

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