I know that we can get $P(A|B,C) = P(A|C)$ if $A$ and $B$ are conditionally independent given $C$. I just wonder that that if we have $P(A|B,C) = P(A|C)$, can we conclude that $A$ and $B$ are conditionally independent given $C$? If so, how to prove it?
2026-05-06 00:02:23.1778025743
Conditional Independence: $P(A|B,C) = P(A|C) \to A$ and $B$ are conditionally independent given $C$
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Show that the statement $\mathsf P(A\mid B,C)=\mathsf P(A\mid C)$ is equivalent to $\mathsf P(A,B\mid C)=\mathsf P(A\mid C)~\mathsf P(B\mid C)$ by means of the definition of conditional probability.
$$\begin{align}\mathsf P(A\mid B,C)&=\mathsf P(A\mid C)\\&~~\vdots\\&~~\vdots\\\dfrac{\mathsf P(A,B\mid C)}{\mathsf P(B\mid C)}&=\mathsf P(A\mid C)\\\therefore\qquad\mathsf P(A,B\mid C)&=\mathsf P(A\mid C)~\mathsf P(B\mid C)\end{align}$$