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1 PREVIEW OF THE BOOK 7 information I (X; Y ) = H (X) − H (X|Y ) = p(x, y) log x,y p(x, y) . 4) The mutual information I (X; Y ) is a measure of the dependence between the two random variables. It is symmetric in X and Y and always nonnegative and is equal to zero if and only if X and Y are independent. A communication channel is a system in which the output depends probabilistically on its input. It is characterized by a probability transition matrix p(y|x) that determines the conditional distribution of the output given the input.

Interpreting this in terms of the data-processing inequality, this implies that θ → T (X) → U (X) → X. 128) Hence, a minimal sufficient statistic maximally compresses the information about θ in the sample. Other sufficient statistics may contain additional irrelevant information. For example, for a normal distribution with mean θ , the pair of functions giving the mean of all odd samples and the mean of all even samples is a sufficient statistic, but not a minimal sufficient statistic. In the preceding examples, the sufficient statistics are also minimal.

16) x∈X y∈Y =− x∈X y∈Y p(x, y) log p(x) − =− x∈X y∈Y p(x, y) log p(y|x) x∈X y∈Y p(x) log p(x) − =− x∈X = H (X) + H (Y |X). 20) and take the expectation of both sides of the equation to obtain the theorem. Corollary H (X, Y |Z) = H (X|Z) + H (Y |X, Z). 21) Proof: The proof follows along the same lines as the theorem. 1 Let (X, Y ) have the following joint distribution: The marginal distribution of X is ( 12 , 14 , 18 , 18 ) and the marginal distribution of Y is ( 14 , 14 , 14 , 14 ), and hence H (X) = 74 bits and H (Y ) = 2 bits.

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Advances in Computers, Vol. 21


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