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Given P(A|B) and P(A|C), how to get or strategically approach P(A|(B & C))?

Is there a way to approach this if it is not known whether B and C are dependent? If not, how to get P(A|(B & C)) assuming B and C are independent, or, dependent?

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    Try expanding P(A,B,C) using the law of total probability in different ways.2011-04-26
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    From a purely mathematical viewpoint, I don't think there's a way to analytically derive one from the other. However, since you say "strategically", I'll link to a fairly well-known (but ugly) hack in machine learning circles called product of experts: http://www.cs.toronto.edu/~hinton/absps/nccd.pdf. It makes the assumption that $p(A|B,C) \propto P(A|B)P(A|C)$ and lives with it. It's utterly wrong, but it works for engineering purposes.2011-04-26

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