Processing math: 61%
Google
 
Web unafbapune.blogspot.com

Friday, January 10, 2014

 

Proposition 2.7 Markov Chain

For random variables X1,X2,,Xn, where n3,X1X2Xn forms a Markov chain if
  p(x1,x2,,xn)={p(x1,x2)p(x3|x2)p(xn|xn1)if p(x2),p(x3),,p(xn1)>00otherwise.
Show that X1X2Xn forms a Markov Chain if and only if XnXn1X1 forms a Markov Chain.

By definition,
  p(a|b)=p(a,b)p(b)
Furthermore, if X1X2Xn forms a Markov Chain,
  p(x1,x2,,xn)=p(x1,x2)p(x3|x2)p(xn1|xn2)p(xn|xn1)=p(x1,x2)p(x3,x2)p(x2)p(xn1,xn2)p(xn2)p(xn,xn1)p(xn1)=p(x1,x2)p(x2)p(x3,x2)p(x3)p(xn2,xn1)p(xn1)p(xn,xn1)=p(xn,xn1)p(xn2|xn1)p(x2|x3)p(x1|x2)
forms a Markov Chain.

\Box

Source: Information Theory.


Comments: Post a Comment

<< Home

This page is powered by Blogger. Isn't yours?