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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 →∞,

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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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