1
$\begingroup$

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.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$
    – william_grisaitis
    Commented 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
    Commented Apr 10, 2020 at 7:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.