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Prove that conditional independence is symmetric (i.e. if A is independent of B given C then...

Prove that conditional independence is symmetric (i.e. if A is independent of B given C then B is independent of A given C).

Please type the answer for my notes thanks!

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Answer #1

Given, A is independent of B given C.

i.e., P(A|B, C) = P(A|C)----------------(1)

Now, P(B|A, C) = P(BA|C) / P(A|C)--------------(2)

From equation (1), using the value of P(A|C) into equation (2), we get-

P(B|A, C) = P(BA|C) / P(A|B, C)

P(B|A, C)*P(A|B, C) = P(BA|C)-----------(3)

From (3),

B is independent of A given C. ( If two events E and F are independent then P(E)*P(F) = P(EE) )

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