Explicit joint probability distribution.

Click For Summary

Discussion Overview

The discussion revolves around the computation of the joint probability distribution P(x,y,z) given the marginal and conditional probabilities P(x), P(y|x), and P(z|x). Participants explore the implications of independence between the variables Y and Z in the context of probability theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions how to compute P(z|x,y) with the given data and suggests that since Y does not depend on Z directly, P(z|x,y) might equal P(z|x).
  • Another participant asks whether Y and Z are independent of each other.
  • A subsequent post reiterates the question of independence and notes that the problem comes from a book where P(y|z) and P(z|y) are not provided, leading to the assumption of independence.
  • It is proposed that if Y and Z are independent, then P(z|x,y) simplifies to P(z|x).
  • Another participant agrees that the joint probability distribution would then be expressed as P(x,y,z) = P(x)P(y|x)P(z|x), suggesting a symmetry in the causal network of Y and Z.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the independence of Y and Z, with some assuming independence based on the lack of certain conditional probabilities. The discussion remains unresolved as to whether the assumption of independence is valid.

Contextual Notes

The discussion highlights the dependence on assumptions regarding independence and the implications of those assumptions on the computation of joint probabilities. There is no consensus on the independence of Y and Z.

carllacan
Messages
272
Reaction score
3
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.
 
Physics news on Phys.org
Are y and z independent of each other?
 
Stephen Tashi said:
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.
 
Last edited:
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))
 
  • Like
Likes   Reactions: 1 person
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.
 

Similar threads

Undergrad Expected value of X~Geom(p) given X+Y=z
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Help understanding conditional expectation identity
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Understanding extinction probability in simple GWP
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Linear regression and random variables
  • · Replies 30 ·
2
Replies
30
Views
5K
MHB Distribution, expected value, variance, covariance and correlation
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad How to obtain moment bound from the importance sampling identity?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Probability - Amount of money in pocket
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Write probability in terms of shape parameters of beta distribution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Find P(X+Y>1/2) for given joint density function
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Is this conditional expectation identity true?
  • · Replies 1 ·
Replies
1
Views
2K