# If A and B are independent, can P(C | A, B) be expressed only in terms of P(A), P(B), P(C | A), and P(C | B)?

Conditional probability question.

Let's say I have...

• three random variables: A, B, C
• A and B are independent
• C depends on A and B

My question is: can I express P(C | A, B) in terms of P(A), P(B), P(C | A), and P(C | B)?

Asking this because I'm studying Bayesian networks, and I'm wondering if it's possible to define a node's "complete" conditional probability distribution (i.e. P(X | all parents of X)) given only "partially conditional" distributions, like one conditional per parent ({P(X | parent 1), P(X | parent 2), ...}).

I have a hunch that this is not possible. I think we also need to know P(A, B | C). Just struggling to prove this analytically.

If you had known $$P(A|B,C)$$, then you could calculate $$P(C|A,B)$$ as follows:
$$P(C|A,B)=\frac{P(A|B,C)P(B,C)}{P(A,B)}$$
You already know the joint probabilities $$P(A,B)$$ and $$P(B,C)$$ with what you have.
Due to the usage of Bayes Rule, without a three-term (e.g. involving $$A,B,C$$ together) probability expression, it doesn't make sense to calculate another three-term probability expression.
• @grisaitis pairwise independences between two variables doesn't mean conditional independence given some other, so you cannot reduce a triple term expression into double term expression. You need to have something like $A$ and $C$ are independent given $B$ to reduce $P(A|B,C)$ to $P(A|B)$ for example. Apr 10, 2020 at 7:52