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i am sure this is quite stupid and i know i am doing something wrong here but cannot figure out what. consider example below, assume b and c are independent; a is dependent on b and c: p(a) = .3, p(~a) = .7 assume p(b|a) = .8, p(b|~a) = .3, so p(b) = .8X.3+.3X.7 = .45 assume p(c|a) = .9, p(c|~a) = .1, so p(c) = .9X.3+.1X.7 = .34 now p(b&c|a) = .8X.9 = .72, p(b&c|~a) = .3X.1 = .03, so p(b&c) = .72X.3+.03X.7=.237 so far so good, the problem is i can also calculate p(b&c) as: p(b&c) = p(b)Xp(c) = .45X.34 = .153 now i got 2 answers for p(b&c), .237 & .153. why the discrepancy? thanks! |
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Short answer: The discrepancy is because, in your example, B and C are not independent. They're independent given A, which is different. Long answer: Viewed as a Bayesian network, the independence description you give corresponds to the following BN: B -> A <- C Which leads to the following factorization: P(A,B,C) = P(B) P(C) P(A|B,C) However, your example was equivalent to the following BN: B <- A -> C Which leads to a different factorization: P(A,B,C) = P(A) P(B|A) P(C|A) When A is unknown, B and C are not independent. For more information, read any textbook about Bayesian networks and note the comments about d-separation. exactly what i needed to break the jam. thanks!
(Jul 13 '10 at 22:31)
tx42
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