By Raymond W. Yeung (auth.)

A First path in details concept is an updated advent to info idea. as well as the classical subject matters mentioned, it offers the 1st accomplished therapy of the idea of I-Measure, community coding conception, Shannon and non-Shannon sort details inequalities, and a relation among entropy and staff idea. ITIP, a software program package deal for proving info inequalities, can also be incorporated. With a great number of examples, illustrations, and unique difficulties, this e-book is great as a textbook or reference ebook for a senior or graduate point path at the topic, in addition to a reference for researchers in comparable fields.

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In the rest of the section , we will show that stationarity is a sufficient condition for the existence of the entropy rate of an information source. 54 Let {Xk} be a stationary source. Then H'x exists. LEMMA Proof Since H(XnIX 1 , X 2, " ', X n- 1 ) is lower bounded by zero for all n, it suffices to prove that H(XnIX1,X2 " " ,Xn- d is non-increasing in n to conclude that the limit H'x exists. Toward this end, for n 2: 2, consider H(XnIX 1,X2,'" , X n- d ::; H(X nIX2, X3, " ' , X n- 1 ) = H(Xn-dX 1 , X2," ', X n- 2), where the last step is justified by the stationarity of proved.

Thus alI Shannon's information measures are special cases of conditional mutual information. Therefore, we only need to discuss the continuity of conditional mutual information. 66) PY where we have written I(X; YIZ) and SXYZ as Ip(X; YIZ) and SXYz(P), respectively to emphasize their dependence on p. Since log a is continuous Information Measures 17 > 0, Ip(X; Y IZ) varies continuously with p as long as the support S Xy Z(p) does not change. The problem arises when some positive probability in a for a masses become zero or some zero probability masses become positive.

Xl, X 2," ', X n are mutually independent. 130) -t Y -t Z forms a Ma rkov chain. Proof By the chain rule for mutual information, we have I (X; Y, Z) = I(X ; Y) + I(X ; ZIY) ~ I(X ; Y). The above inequality is tight if and only if I(X j ZIY) forms a Markov chain. The theorem is proved. 132) I(X j Z) ~ I(Y j Z). 133) and Before proving this inequality we discuss its meaning. Suppose X is a random variable we are interested in, and Y is an observation of X. If we infer X via Y , our uncertainty about X on the average is H(XIY) .