$P(A | B,C) = P(A | C)$ means that A and B are independent conditioned on C, that is, if C is given, then A and B are independent, otherwise, they may not be. How do I prove this?
2026-02-22 21:00:13.1771794013
How do I prove A and B are independent given C?
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We have $$ P(A\mid B,C) = \frac{P(A,B,C)}{P(B,C)} = \frac{P(A,B,C)}{P(B\mid C)P(C)} = \frac{P(A,B\mid C)}{P(B\mid C)}$$ so $$ P(A\mid B,C) = P(A\mid C) \iff\frac{P(A,B\mid C)}{P(B\mid C)} = P(A\mid C) \iff P(A,B\mid C) = P(A\mid C)P(B\mid C)$$ which is probablty the statement of "$A$ and $B$ are conditionally independent given $C$" you had in mind.