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Explicit joint probability distribution.

  1. May 30, 2014 #1
    Hi!

    Suppose we have two variables Y and Z that depend on a third one, X. We are given P(x), P(y|x) and P(z|x). The joint probability distribution P(x,y,z), according to the chain probability rule, is given by P(x,y,z) = P(x)P(y|x)P(z|x,y)

    But how can we compute P(z|x,y) with the given data?

    Since Y does not depend on Z directly I "feel" that P(z|x,y) = P(z|x)(Px) but I can't find a logical reason for it.

    Can you lend me a hand?

    Thank you for your time.
     
  2. jcsd
  3. May 30, 2014 #2

    Stephen Tashi

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    Are y and z independent of each other?
     
  4. May 30, 2014 #3
    This actually comes from an exercise in a book and neither P(y|z) nor P(z|y) are given, so I assume so.
     
    Last edited: May 30, 2014
  5. May 30, 2014 #4

    HallsofIvy

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    If z and y are independent then P(z|x,y) is just P(z|x). (NOT "P(z|x)P(x)" which is P(z))
     
  6. May 30, 2014 #5
    And the joint probability distribution would simply be P(x,y,z) = P(x)P(y|x)P(z|x), right?

    It which makes sense because Y and Z are, so to speak, symmetrical in the causal network, so they should also be symmetrical in this expression.
     
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