Download A First Course in Information Theory by Raymond W. Yeung (auth.) PDF

By Raymond W. Yeung (auth.)

A First direction in info conception is an up to date creation to info conception. as well as the classical subject matters mentioned, it offers the 1st complete therapy of the speculation of I-Measure, community coding conception, Shannon and non-Shannon variety details inequalities, and a relation among entropy and team thought. ITIP, a software program package deal for proving details inequalities, is usually incorporated. With loads of examples, illustrations, and unique difficulties, this ebook is superb as a textbook or reference booklet for a senior or graduate point direction at the topic, in addition to a reference for researchers in comparable fields.

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18. Pinsker's inequality Let d(p, q) denotes the variational distance between two probability distributions p and q on a common alphabet X. We will determine the largest c which satisfies D(pllq) ~ cd2(p, q). a) Let A = {x : p(x) ~ q(x)} , p = {p(A) , 1 - p(A)} , and ij {q(A), 1- q(A)} . Show that D(pllq) ~ D(pllij) and d(p,q) = d(p, ij). b) Show that toward determining the largest value of c, we only have to consider the case when X is binary. c) By virtue of b), it suffices to determine the largest c such that p I-p q 1- q p log - + (1 - p) log - - - 4c(p - q)2 ~ 0 for all 0 ~ p , q ~ 1, with the convention that 0 log % = 0 for b ~ 0 and a log IT = 00 for a > O.

38) x, y For two random variables, we define in the following the conditional entropy of one random variable when the other random variable is given. 15 For random variables X and Y, the conditional entropy of Y given X is defined by H(YIX) = - LP(x, y) logp(yl x) = -Elogp(YIX). 39), we can write H(YIX) = ~p(x) [- ~p(Ylx) IOgp(Y1 X)] . 40) The inner sum is the entropy of Y conditioning on a fixed x E Sx. x) logp(ylx) . 43) z where H(YIX,Z = z) = - LP(x,Ylz)logp(Ylx,z). 45) H(X, Y) = H(Y) + H(XIY) .

I98) The first term tends to 0 as n --+ 00 . Therefore, for any E > 0, by taking n to be sufficiently large, we can make Ibn - al < 2E. Hence bn --+ a as n --+ 00 , proving the lemma . 0 We now prove that H'x is an alternative definition/interpretation of the entropy rate of {Xd when {Xd is stationary. 56 For a stationary source {Xk} . the entropy rate HXt exists, and it is equal to H'x . 54 that H'x always exists for a stationary source {Xd, in order to prove the theorem, we only have to prove that H x = H'x .

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