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Additional info for Amorphous Semiconductors, 2nd Edition (Topics in applied physics, vol. 36)
Rajski (1961); for a simple proof cf.
Familiarity with a few identities and inequalities will make such computations perspicuous. Also, these formal properties of information measures have intuitive interpretations which help remembering and properly using them. Z, . . be RVs with finite ranges X, Y, Z, . . We shall consistently use the notational convention introduced in Section 1 that information quantities featuring a collection of RVs in the role of a single RV will be written without putting this collection into brackets. We shall often use a notation explicitly bringing out that information measures associated with RVs are actually determined by their (j° m t ) distribution.
Xn) also form a Markov chain. The same holds for conditional Markov chains, too. O §3. FORMAL PROPERTIES OF SHANNON'S INFORMATION MEASURES Proof I(X! 55 It suffices to prove the first assertion. We show that Xj_ ! ,ΛΧ^,Ι^. D In a Markov chain, any two RVs depend only through the intermediate ones, hence intuition suggests that their mutual information cannot exceed that of two intermediate RVs. A related phenomenon for hypothesis testing is that one cannot gain more information for discriminating between the hypotheses P and Q when observing the outcome of the experiment with less accuracy, possibly subject to random errors.