What is the relationship between $P(A|B,C) \text{ and } P(A|B)$, such that $P(B,C)$ is joint distribution? Using product rule, I got so far as to $$P(A|B,C) = \frac{P(A, B, C)}{P(B,C)}$$, but I do not know how to continue from there.
2026-04-07 02:13:26.1775528006
Separate joint distribution from conditional joint distribution
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You have that: $\def\P{\operatorname{\rm P}}~~~\P(A,B,C) =\P(B,C)\P(A\mid B,C)$
It is also so that: $\P(A,B,C)=\P(B)\P(A\mid B)\P(C\mid A,B)$
Clearly, therefore :$$\dfrac{\P(B,C)\P(A\mid B,C)}{\P(A\mid B)}=\P(B)\P(C\mid A,B)$$