0
$\begingroup$

$P(A,B,C) = P(A)P(B)P(C|A,B)$. I understand how $A,B$ are marginally independent on $C$, but I'm confused as to how the $A, B$ are conditionally dependent on $C$.

$P(A,B|C) = \frac{P(A)P(B)P(C|A,B)}{P(C)}$, and $P(C)$ is the marginal of $C$ over $P(A)P(B)P(C|A,B)$, but that was as far as I got.

$\endgroup$
0
$\begingroup$

If they would be conditionally independent, it would be possible to decompose $$P(A,B|C)=P(A|C)P(B|C)\textrm{.}$$ However, as you correctly write $$P(A,B|C) = \frac{P(A)P(B)P(C|A,B)}{P(C)}\textrm{.}$$ Thus, whether $A$ and $B$ are conditionally dependent depends on the decomposition of $P(C|A,B)=Q_1(A,C)Q_2(B,C)$ where $Q_1$ and $Q_2$ are some nonnegative functions. If they exist, $A$ and $B$ are conditionally independent for given $C$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.