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Conditional Indepedence

Eustice_Langham

Active Member
There is a section within the GARP notes on page 6 of the Quantitative Analysis book that discusses Conditional Independence and I hope that someone may be able to explain this concept is simpler terms that what is shown in the notes.

Particularly this section..."Note that two types of independence—unconditional and conditional—do not imply each other. Events can be both unconditionally dependent (i.e., not independent) and conditionally independent. Similarly, events can be unconditionally independent, yet conditional on another event they may be dependent."..

Thanks
 

lushukai

Member
Subscriber
Dear @Eustice_Langham ,

I took reference from a few YouTube videos and Wiki pages (you can search conditional independence and conditional dependence), there seems to be four combinations we are talking about:

Quote Pt 1: "Note that two types of independence—unconditional and conditional—do not imply each other …"

1. Unconditional Independence
P(A ∩ B) = P(A) * P(B)

2. Conditional Independence
P(A, B | C) = P(A | C) * P(B | C)

Answer for Quote Pt 1: Note that (1.) or (2.) cannot be used to deduce the other.

Quote Pt 2: "... Events can be both unconditionally dependent (i.e., not independent) and conditionally independent. Similarly, events can be unconditionally independent, yet conditional on another event they may be dependent."

3. Unconditional Dependence
You can break this down into 3.1 Unconditional Positive Dependence and 3.2 Unconditional Negative Dependence. For
3.1 P(A | B) > P(A) or P(B | A) > P(B)
and
3.2 P(A | B) < P(A) or P(B | A) < P(B)
You can think of A and B happening outside a third probability space C (with reference to 4.).

4. Conditional Dependence
Wikipedia states that you can break (4.) into 4.1 Conditional Positive Dependence and 4.2 Conditional Negative Dependence. For
4.1 P(A | B, C) > P(A | B) or P(B | A, C) > P(B | A)
and
4.2 P(A | B, C) < P(A | B) or P(B | A, C) < P(B | A)
Let us take the example of the left formula for (4.2). What it is trying to say is that, the probability of A occurring given that B has occurred in the probability space of C is less than the probability of A occurring given that B has occurred outside the probability space of C. You can use this explanation for the formulas in 4.1 as well.

Answer for Quote Pt 2: For events that are unconditionally dependent and conditionally independent, you can think of it as A or B having higher or lower probability (positive/negative dependence) of occurring outside the probability space of C given that the other has occurred (unconditionally dependent) and A or B having its own probability of occurring inside the probability space of C regardless of whether other has occurred (conditionally independent).
You can use this logic to deduce the meaning of the second part of Quote Pt 2.

My apologies if the explanation is a bit theoretical or unclear. You can also refer to this thread that David has commented on: https://www.bionicturtle.com/forum/threads/probability-conditional-independence.23320/
 
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