Conditional Independence and Independence question

R, this conversation is discussing the relationship between two variables, T and C, given a third variable Z. The question asks if T and C are independent given Z, does it follow that they are also independent without Z. The answer is no, as the independence of T and C depends on the value of Z. Similarly, if T and C are independent without Z, it does not necessarily mean they are also independent given Z. In summary, the independence of T and C is not guaranteed in either case.
  • #1
akolman
1
0
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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  • #2
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
 
Last edited:

Related to Conditional Independence and Independence question

1. What is conditional independence?

Conditional independence is a statistical concept that refers to the relationship between two variables. It means that the value of one variable does not affect the probability of the other variable occurring, given the value of a third variable.

2. How is conditional independence different from regular independence?

Conditional independence is a more specific concept than regular independence. Regular independence means that the occurrence of one variable does not affect the probability of another variable occurring. Conditional independence takes into account the influence of a third variable on the relationship between the two variables.

3. How is conditional independence determined?

Conditional independence is determined by analyzing data and calculating the conditional probability of two variables given a third variable. If the conditional probability is equal to the regular probability, then the two variables are considered conditionally independent.

4. What are some real-world examples of conditional independence?

One example of conditional independence is the relationship between smoking and lung cancer. While smoking is a major risk factor for lung cancer, it may not have the same influence on the development of lung cancer in individuals who have a genetic predisposition to the disease. In this case, smoking and lung cancer are conditionally independent variables, with the third variable being genetic predisposition.

5. How is conditional independence useful in scientific research?

Conditional independence is useful in scientific research because it allows researchers to better understand and analyze the relationship between variables. By taking into account the influence of a third variable, researchers can make more accurate predictions and draw more meaningful conclusions from their data.

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