What does conditional independence mean?

• A. A and B are independent regardless of any conditioning.
• B. A and B are always mutually exclusive.
• C. A and B are independent given C if P(A ∩ B) = P(A)P(B).
• D. A and B are conditionally independent given C if P(A ∩ B | C) = P(A|C) P(B|C).

Answer: (D)

Conditional independence means that once you know the value of C, A and B behave as if they are unrelated. In symbols, the joint probability of A and B given C factors into the product of their individual probabilities given C: P(A ∩ B | C) = P(A | C) · P(B | C). Put differently, knowing B doesn’t change the likelihood of A when C is known, and vice versa.

This is why the statement is correct: it directly encodes that A and B are independent within each level of C by using conditional probabilities. It captures the idea that C accounts for any association between A and B; after conditioning on C, A and B no longer inform each other.

The other ideas don’t fit because they describe different notions. Independence without any conditioning means A and B are independent in the overall, unconditional sense, which is not the same as independence given C. Mutual exclusivity would require that A and B cannot occur together, which is stronger and not what conditional independence asserts. And using P(A ∩ B) without conditioning describes unconditional independence, not conditional independence; the condition must be inside the probability to reflect dependence only through C.