By Raymond W. Yeung (auth.)

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

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33) that H(X) ~ O. 41) that H(YIX) ~ O. 35 H(X) = 0 if and only if X is deterministic. , there exists x * E X such that p(x*) = 1 and p(x) = 0 for all x =I x *, then H(X) = -p(x*) logp(x*) = O. , there exists x* E X such that 0< p( x*) < 1, then H(X) ~ -p( x*) logp(x*) > O. Therefore, we conclude that H(X) = 0 if and only if X is deterministic. 36 H(YIX) = 0 if and only ifY is afunction of X . 41), we see that H(YIX) = 0 ifand only if H(YIX = x ) = 0 for each xESx - Then from the last proposition, this happens if and only if Y is deterministic for each given x.

Then f'{a) = Y]« - 1 and f"{a) = -1/a 2 . Since f(l) = 0,1'(1) = 0, and 1"(1) = -1 < 0, we see that f(a) attains its maximum value 0 when a = 1. 88) . 88) if and only if a = 1. 3 is an illustration of the fundamental inequality. 89) 1. 3 that the fundamental inequality results from the convexity of the logarithmic function . In fact, many important results in 21 Information Measures information theory are also direct or indirect consequences of the con vexit y of the logarithmic functi on ! 90) if and only if p = q.

98) . Equality holds if and only if a~ constant for all i. The theorem is proved. 0 = b~ for all i, or ~ = One can also prove the divergence inequality by using the log-sum inequality (see Problem 15), so the two inequalities are in fact equivalent. The logsum inequality also finds application in proving the next theorem which gives a lower bound on the divergence between two probability distributions on a common alphabet in terms of the variational distance between them. 32 Let p and q be two probability distributions on a common alphabet X .