# Conditional Independence

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!

So I know that $$C$$ and $$D$$ are independent given $$P(A,B)$$ right?
They are independent given $$A,B$$, not $$P(A,B)$$.
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: $$P(A,B,C,D) = P(A,B) \cdot P(C|A) \cdot P(D|B)$$?