MHB Conditional Independence and Independence question

AI Thread Summary
The discussion revolves around the concepts of conditional independence and independence in probability theory. The user poses two questions regarding whether conditional independence of T and C given Z implies their unconditional independence, and vice versa. It is suggested that neither implication necessarily holds true, as the relationship between T and C can be defined independently when Z is not considered. The user seeks clarification on how to demonstrate these points mathematically. Understanding these nuances is crucial for accurately applying concepts of independence in statistical analysis.
akolman
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Hello, I am stuck with the following question.

1. Suppose T ind. C |Z, does it follow that T ind. C ?

2. Suppose T ind. C , does it follow that T ind. C |Z?

I think both don't follow, but I don't know how to show it

Thanks in advance
 
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akolman said:
Hello, I am stuck with the following question.

1. Suppose T ind. C |Z, does it follow that T ind. C ?

2. Suppose T ind. C , does it follow that T ind. C |Z?

I think both don't follow, but I don't know how to show it

Thanks in advance

\(T\) and \(C\) independent given \(Z\) means:

\(P(T \wedge C|Z)=P(T|Z)P(C|Z)\)

Now we are free to define any relation we want between \(T\) and \(C\) if \(\neg Z\) is the case so that

\(P(T \wedge C) \ne P(T)P(C)\)

CB
 
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