Explicit joint probability distribution.

[carllacan]

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.

 

[Stephen Tashi]

Are y and z independent of each other?

 

[carllacan]

Are y and z independent of each other?

This actually comes from an exercise in a book and neither P(y|z) nor P(z|y) are given, so I assume so.

 

[HallsofIvy]

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))

 

[carllacan]

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.