Show that x and y are independent in this joint distribution

  • Thread starter endeavor
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  • #1
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Given that the joint probability Pr(w,x,y,z) over four variables factorizes as
[itex]Pr(w,x,y,z) = Pr(w) Pr(z|y) Pr(y|x,w)Pr(x)[/itex]
show that x is independent of w by showing that Pr(x,w) = Pr(x)Pr(w).

Attempt: if we simply assume Pr(x,w) = Pr(x)Pr(w), then:
[itex]
\begin{align}
Pr(w,x,y,z) &= Pr(w) Pr(z|y) Pr(y|x,w) Pr(x)\\
&\stackrel{?}= Pr(x,w) Pr(z|y) Pr(y|x,w)\\
&\stackrel{?}= Pr(z|y) Pr(w,x,y)
\end{align}
[/itex]

But can we say the last line equals Pr(w,x,y,z)? I think this problem can be solved by simply applying Bayes' rule several times, but I can't seem to wrap my head around it.
 

Answers and Replies

  • #2
798
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What we need to prove is P(w,x) = P(w)P(x). P(w,x) can be computed easily by integrating (or summing) P(w,x,y,z) over all y and z :wink:
 
  • #3
176
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Ah, I see. So we have:
[itex]
\begin{align}
Pr(x,w) &= \iint Pr(w,x,y,z)\,dydz \\
&= \iint Pr(w)Pr(z|y)Pr(y|x,w)Pr(x)\,dydz \\
&= Pr(w)Pr(x) \iint Pr(z|y)Pr(y|x,w)\,dydz \\
&= Pr(w)Pr(x)
\end{align}
[/itex]
 

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