MHB Conditional Independence and Independence question

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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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