TITLE: Conditional independence, is $P(A|C) = \sum_{B} P(A,B|C)$
QUESTION [0 upvotes]: The following holds true $$P(A) = \sum_{B} P(A,B)$$ if we have a condition does it still hold true
$$P(A|C) = \sum_{B} P(A,B|C)$$

REPLY [1 votes]: Yes, it does.
$$\frac{P(A,C)}{P(C)} = \sum_B \frac{P(A,B,C)}{P(C)}$$
$$P(A,C)=\sum_BP(A,B,C)$$
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