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Additional info for Advances in Computers, Vol. 21
The nonnegativity of all Shannon’s information measures is called the basic inequalities. For entropy and conditional entropy, we offer the following more direct proof for their nonnegativity. Consider the entropy H(X) of a random variable X. For all x ∈ SX , since 0 < p(x) ≤ 1, log p(x) ≤ 0. 35) that H(X) ≥ 0. 43) that H(Y |X) ≥ 0. 35. H(X) = 0 if and only if X is deterministic. Proof. , there exists x∗ ∈ X such that p(x∗ ) = 1 and p(x) = 0 for all x = x∗ , then H(X) = −p(x∗ ) log p(x∗ ) = 0. , there exists x∗ ∈ X such that 0 < p(x∗ ) < 1, then H(X) ≥ −p(x∗ ) log p(x∗ ) > 0.
The following theorem renders a solution to this problem. 50. 200) are satisfied. 200). Proof. 210) ≥ 0. 200). 31). The proof is accomplished. Remark For all x ∈ S, p∗ (x) > 0, so that Sp∗ = S. 50 is rather subtle. 51. 212) for all x ∈ S. Then p∗ maximizes H(p) over all probability distribution p defined on S, subject to the constraints p∗ (x)ri (x) p(x)ri (x) = x∈Sp for 1 ≤ i ≤ m. 52. 200) be empty. 214) a constant that does not depend on x. , p∗ (x) = |S|−1 for all x ∈ S. 43. 53. , the mean of the distribution p is fixed at some nonnegative value a.
249) that H(X(m)) → H(X) as m → ∞. 255) Chapter Summary 43 Chapter Summary Markov Chain: X → Y → Z forms a Markov chain if and only if p(x, y, z) = a(x, y)b(y, z) for all x, y, and z such that p(y) > 0. Shannon’s Information Measures: H(X) = − p(x) log p(x) = −E log p(X) x I(X; Y ) = p(x, y) log x,y H(Y |X) = − p(x, y) p(X, Y ) = E log p(x)p(y) p(X)p(Y ) p(x, y) log p(y|x) = −E log p(Y |X) x,y I(X; Y |Z) = p(x, y, z) log x,y,z p(X, Y |Z) p(x, y|z) = E log . p(x|z)p(y|z) p(X|Z)p(Y |Z) Some Useful Identitites: H(X) = I(X; X) H(Y |X) = H(X, Y ) − H(X) I(X; Y ) = H(X) − H(X|Y ) I(X; Y |Z) = H(X|Z) − H(X|Y, Z).
Advances in Computers, Vol. 21