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I have a joint probability, which factors as follows:

$P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$

So I know that $C$ and $D$ are independent given $P(A, B)$ right?

I want to infer $P(A,B|C,D)$.

I use Bayes' Rule:

$P(A,B,C,D) = P(A,B) \frac{P(A|C) \cdot P(C) \cdot P(B|D) \cdot P(D)}{P(A) \cdot P(B)}$

Then I condition on $P(C,D)$:

$P(A,B|C,D) = P(A,B) \frac{P(A|C) \cdot P(C) \cdot P(B|D) \cdot P(D)}{P(A) \cdot P(B) \cdot P(C,D)}$

My problem is, I do not know the terms $P(C)$ and $P(D)$ nor $P(C,D)$. What additional assumption would I have to make so that I can write:

$P(A,B|C,D) = P(A,B) \frac{P(A|C) \cdot P(B|D)}{P(A) \cdot P(B)}$

or

$P(A,B|C,D) \propto P(A,B) \frac{P(A|C) \cdot P(B|D) }{P(A) \cdot P(B)}$

Would it help if I assume, that both $P(A)$ and $P(B)$ are equally distributed? I guess, that then I at least can write the latter one, right?

Any help is very much appreciated! Thank you very much!

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So I know that $C$ and $D$ are independent given $P(A,B)$ right?

They are independent given $A,B$, not $P(A,B)$.

I use Bayes' Rule: ...

Generally, you use Bayes' rule to write something in terms of things you do know. Why not start with the equation you wrote before that: $P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$?

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