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Show that x and y are independent in this joint distribution

  1. Jan 31, 2013 #1
    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.
     
  2. jcsd
  3. Jan 31, 2013 #2
    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:
     
  4. Feb 1, 2013 #3
    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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