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Consider I have three random variables A, B, C. I know that A depends on (B,C). Can I always deduce that it implies that A depends on B and also A depends on C? I mean does it implies that neither A is independent of B or A is independent of C?

I tried to find my answer using the definition that if A and (B, C) are independent then $p(A| B, C)= p(A)$ therefore, $\frac{p(A\cap (B, C))}{p(B, C)} = p(A)$. Then, if I want to use proof by contradiction, I can suppose that A and B are independent. therefore, $p(A|B)= \frac{p(A \cap B)}{p(B)}=p(A)$. But I cannot continue my proof from this part, since for the equation of $\frac{p(A\cap (B, C))}{p(B, C)} = p(A)$ I need something about $p(B, C)$ not, A and B.

I also know that A and B and C are independent if they all mutually be independent, and A be independent of (B, C), B be independent of (A, C), and C be independent of (A, B).

Based on this, is this deduction correct:

If I know that A is independent of (B, C), we must have

  1. A be independent of B, and

  2. A be independent of C.

Therefore, if one of these two conditions doesn't hold, then A is never independent of (B, C) jointly. Thus,

when I know that A depends on (B, C), I may have A is independent of B.

Is this sentence correct?

I got confused. How is that possible? the fig (d) in this answer shows that in v structure, Z depends on (X, Y) and also Z depends on X and Z depends on Y. I got confused. How can I imagine somehow the same figure when Z depends on (X, Y) and also Z depends on X, but Z is independent of Y?

(Sorry if my notation is not precise in statistics)

utobi
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m123
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  • When $A$ depends on $B,$ then a fortiori $A$ depends on $(B,C),$ no matter what the relationship between $A$ and $B$ might be. Your question appears to be a matter of incorrect logic: when a proposition $P$ implies $Q,$ it doesn't follow that the negation of $P$ implies the negation of $Q.$ – whuber Oct 06 '23 at 20:14
  • It's important to keep in mind that it's possible for three variables to be pairwise independent but not jointly independent (not mutually independent). Site searches turn up multiple posts on that issue, including some with explicit examples. – Glen_b Oct 07 '23 at 01:15
  • @Glen_b: I know about independence, but I got confused by "dependence" not independent (I know they are opposite, but in usage I get confused). Suppose I have the information that my three variables are jointly dependent. Can I deduce that they are pairwise dependent too? – m123 Oct 07 '23 at 18:06
  • @whuber: By $P\rightarrow Q$, do you mean: 1. if three variables are jointly independent, then they are pairwise independent. or 2. if three variables are pairwise independent, (and also A with (B, C), and B with (A, C), and C with (A, B), then they are jointly independent? 1? or 2 or both? which one is the correct statement? – m123 Oct 07 '23 at 18:12
  • I don't know, because I haven't used that right arrow symbol here. – whuber Oct 07 '23 at 19:46
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    @m123 No. It's covered by applying the rule of transposition to my statement. i.e. what I said implies "no" to what you asked about. You will almost certainly have used this rule in ordinary reasoning in your life constantly -- multiple times a day, I'd expect. Perhaps you've just never noticed that you do it. "If it's raining, Joan always carries an umbrella. Joan doesn't have an umbrella, so it's not raining". That's the form of reasoning you apply to my statement to derive that they're not necessarily pairwise dependent. – Glen_b Oct 07 '23 at 21:28
  • See also modus tollens – Glen_b Oct 07 '23 at 22:46

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