I am studying conditional and joint probabilities, and a particular thought has me stumped. In a scenario with three different variables with binary outcomes, A, B, C and no independence or dependence assumptions, is the following always true?
P(A|B,C)*P(B|A,C)*P(C|B,C) = P(A,B,C)
My first assumption was no because I was aware that P(A|B,C)*P(B|C)*P(C) = P(A,B,C) regardless of independence assumptions. I thought adding in these addition dependences in the equation in question would mess things up and make the equivalency false.
Then I started thinking of some scenarios. If A, B, and C were all independent, then P(A|B,C)*P(B|A,C)*P(C|B,C) would reduce to P(A)*P(B)*P(C) and be equivalent to P(A,B,C). Further, if you start with the expression P(A|B,C)*P(B|A,C)*P(C|B,C) and then, one by one, reduce it according to any independences, you would come to a more concise form of the expression, but it would still be equivalent to P(A,B,C). Is this the case?
TLDR: is P(A|B,C)*P(B|A,C)*P(C|B,C) = P(A,B,C) true regardless of any independence assumptions?