If two or more variables A, B, C, etc. are jointly mutually independent of one another, does this imply that that they are also conditionally independent given some set of conditioning variables X, Y, Z, etc.?
If you test variables A, B, C,, etc., and the hypothesis of independence is not rejected, are you generally safe in assuming that these variables will also survive a test of conditional independence given X, Y, Z, etc.? Is it safe enough that you do not need to bother running the latter test? Why or why not?
Best Answer
No.
Consider three boolean variables: A, B, X where X and A are i.i.d. Bernoulli with probabilty 0.5, while B = X $\oplus$ A (that is, B is equal to the xor of X and A).
It's easy to show that B is also Bernoulli distributed with probabilty 0.5, and A and B are mutually independent, though obviously they aren't conditionally independent given X.