Friday, January 10, 2014
Proposition 2.7 Markov Chain
For random variables X1,X2,⋯,Xn, where n≥3,X1→X2→⋯→Xn forms a Markov chain if
p(x1,x2,⋯,xn)={p(x1,x2)p(x3|x2)⋯p(xn|xn−1)if p(x2),p(x3),⋯,p(xn−1)>00otherwise. |
By definition,
p(a|b)=p(a,b)p(b) |
p(x1,x2,⋯,xn)=p(x1,x2)p(x3|x2)⋯p(xn−1|xn−2)p(xn|xn−1)=p(x1,x2)p(x3,x2)p(x2)⋯p(xn−1,xn−2)p(xn−2)p(xn,xn−1)p(xn−1)=p(x1,x2)p(x2)p(x3,x2)p(x3)⋯p(xn−2,xn−1)p(xn−1)p(xn,xn−1)=p(xn,xn−1)p(xn−2|xn−1)⋯p(x2|x3)p(x1|x2) |
\Box
Source: Information Theory.