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Conditional entropy. Let A and B = (Bo, B₁,..., Bn) be a discrete random variable and vector, respectively. The conditional entropy of A with respect to B is defined as H(A | B) = E (E{-log f(A | B) | B}) where f(a | b) = P(A = a | B = b). Let X be an aperiodic Markov chain on a finite state space. Show that and that H(Xn+1 | X0, X₁. , Xn) = H(Xn+1 | Xn), .... H(Xn+1 | Xn) →→лi Σ Pij i j if X is aperiodic with a unique stationary distribution . log Pij as n →∞,

Conditional entropy. Let A and B = (Bo, B₁,..., Bn) be a discrete random variable and vector, respectively. The conditional entropy of A with respect to B is defined as H(A | B) = E (E{-log f(A | B) | B}) where f(a | b) = P(A = a | B = b). Let X be an aperiodic Markov chain on a finite state space. Show that and that H(Xn+1 | X0, X₁. , Xn) = H(Xn+1 | Xn), .... H(Xn+1 | Xn) →→лi Σ Pij i j if X is aperiodic with a unique stationary distribution . log Pij as n →∞,

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 9EQ
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Conditional entropy. Let A and B = (Bo, B₁,..., Bn) be a discrete random variable and vector, respectively. The conditional entropy of A with respect to B is defined as H(A | B) = E(E{-log f(A | B) | B}) where f(a | b) = P(A = a | B = b). Let X be an aperiodic Markov chain on a finite state space. Show that H(Xn+1 | X0, X₁. , Xn) = H(Xn+1 | Xn), and that H(Xn+1 \ Xn) → - ΣπιΣ pi; i j if X is aperiodic with a unique stationary distribution . log Pij as n →∞,
Transcribed Image Text:Conditional entropy. Let A and B = (Bo, B₁,..., Bn) be a discrete random variable and vector, respectively. The conditional entropy of A with respect to B is defined as H(A | B) = E(E{-log f(A | B) | B}) where f(a | b) = P(A = a | B = b). Let X be an aperiodic Markov chain on a finite state space. Show that H(Xn+1 | X0, X₁. , Xn) = H(Xn+1 | Xn), and that H(Xn+1 \ Xn) → - ΣπιΣ pi; i j if X is aperiodic with a unique stationary distribution . log Pij as n →∞,
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