By Raymond W. Yeung (auth.)

ISBN-10: 1441986081

ISBN-13: 9781441986085

ISBN-10: 1461346452

ISBN-13: 9781461346456

A First path in info thought is an up to date creation to info conception. as well as the classical subject matters mentioned, it offers the 1st finished therapy of the speculation of I-Measure, community coding thought, Shannon and non-Shannon kind info inequalities, and a relation among entropy and crew thought. ITIP, a software program package deal for proving details inequalities, can also be integrated. With quite a few examples, illustrations, and unique difficulties, this e-book is superb as a textbook or reference ebook for a senior or graduate point path at the topic, in addition to a reference for researchers in similar fields.

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However, the existence of the distribution p(Xl , X2 , X3 , X4) constructed immediately after Proposition 2. I2 simply says that it is not always possible to find such a sequence {pd · Therefore, probability distributions which are not strictly positive can be very delicate. 5 that their conditional independence structures are closely related to the factorization problem of such distributions, which has been investigated by Chan and Yeung [43]. 2 SHANNON'S INFORMATION MEASURES We begin this section by introducing the entropy of a random variable.

Let X be a function of Y. Prove that H (X) :S H (Y) . Interpret this result. 12. Prove that for any n ~ 2, n H (X I , X 2," ' , X n ) ~ 2: H(XiIX j , j i= i) . i= 1 13. Prove that Hint: Sum the identities for i = 1,2 , 3 and appl y the result in Problem 12. 14. Let N n = {I , 2, . . , n } and denote H (X i , i E ex) by H (X o. ) for any subset ex of N n . For 1 :S k :S n , let Prove that HI ~ H2 ~ . 39) . See Problem 4 in Chapter 15 for an application of these inequalities. 15. Prove the divergence inequality by using the log-sum inequality.

70) Thus p(x , y, z) logp(z) -+ 0 as p(x , y , z) -+ O. Similarly, we can show that both p(x,y,z)logp(x,z) and p(x ,y,z)logp(y,z) -+ 0 as p(x ,y, z) -+ O. 71) as p(x, y, z) -+ O. Hence, I(X ;YjZ) varies continuously with p even when p(x, y, z) -+ 0 for some x , y, and z . 4 CHAIN RULES In this section, we present a collection of information identities known as the chain rules which are often used in information theory. 23 (CHAIN RULE FOR ENTROPY) n H(X 1,X2 , " ' ,Xn ) = LH(XiIX1 , ,, , ,Xi-d .

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

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