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.


1 Answer 1


If you had known $P(A|B,C)$, then you could calculate $P(C|A,B)$ as follows:


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.

  • $\begingroup$ Thanks! I guess I was wondering, using Bayes rule with a three term expression, if it would be possible to drop one of the conditions due to some independence, and turn a three term expression into a two term expression that would be useful, given the quantities I knew. Oh well. $\endgroup$ Apr 7, 2020 at 23:55
  • 1
    $\begingroup$ @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. $\endgroup$
    – gunes
    Apr 10, 2020 at 7:52

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