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There is 3 random variables, A, B and C I know distribution P(C) and conditional distributions P(A|C) and P(B|C). How to find P(A|B)? I think it's necessary to integrate someone like this:

$$ P(A|B) = \int P((A|B)|C) dP(C) $$

But how to find P((A|B)|C) ?

user1761982
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  • How do you define $P(A|C)$ and $P(B|C)$? – Tom Nov 27 '13 at 21:30
  • @Tom in details, random variable C is an expectation of A and B. A, B, and C are normal-distributed with given dispersions. – user1761982 Nov 27 '13 at 21:37
  • "how to find P((A|B)|C)" Rather, the question would be how to define P((A|B)|C) (this does not exist). – Did Nov 11 '14 at 14:34
  • "I know distribution P(C) and conditional distributions P(A|C) and P(B|C). How to find P(A|B)?" The data (P(C), P(A|C) and P(B|C)) does not determine P(A|B). – Did Nov 11 '14 at 14:36
  • @Tom "How do you define P(A|C) and P(B|C)?" Funny that you would ask because, once one realized that A, B and C are (rather illogically) random variables, this is the only part of the question that causes no problem. – Did Nov 11 '14 at 14:37
  • @Did I suppose it isn't uncommon that if $\nu$ is a measure and $f$ is measurable, you can define $\nu(f) = \int f ,d\nu$.. is that what's happening here? $P(A\mid C) = E_P[A \mid C]$? – Tom Nov 11 '14 at 17:45
  • @Tom My guess is that when X and Y are random variables, the OP writes P(X|Y) for the conditional distribution of X conditioned on Y... but do not start me again on the topic of horrendous notations. :-) – Did Nov 11 '14 at 17:49

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