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Lecture 1: Entropy and mutual information

Tufts UniversityElectrical and Computer EngineeringEE194 Network information TheoryProf. Mai VuLecture 1: Entropy and mutual information1 IntroductionImagine two people Alice and Bob living in Toronto and Boston respectively. Alice (Toronto) goesjogging whenever it is not snowing heavily. Bob (Boston) doesn t ever go that Alice s actions give information about the weather in Toronto. Bob s actions giveno information . This is because Alice s actions are random and correlated with the weather inToronto, whereas Bob s actions are can we quantify the notion of information ?2 EntropyDefinitionTheentropyof a discrete random variableXwith pmfpX(x) isH(X) = xp(x) logp(x) = E[ log(p(x)) ](1)The Entropy measures the expected uncertainty inX.

tions is the relative entropy, also sometimes called the Kullback-Leibler divergence. Definition The relative entropy between two probability distributions p(x) and q(x) is given by D(p(x)||q(x)) = X x p(x)log p(x) q(x). (30) The reason why we are interested in the relative entropy in this section is because it is related

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Transcription of Lecture 1: Entropy and mutual information

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