Combining joint probabilities

Question

Hello,

I am trying to work out the expression for a probability distribution (related to bioinformatics), and am having trouble combining the information about a random variable from two different sources. Essentially, here is the scenario: There are 3 discrete random variables X, A & B. X depends on A and B. A and B are related only through X, i.e. A and B are independent given X. Now, I have derived the expressions for: P(X, A) and P(X, B). I need to calculate P(X, A, B) - this is not a straightforward application of the chain rule.

I can derive P(X | A) from the first expression since P(A) is available. B is never observed independently of A, P(B) is not readily available - at best I can approximate it by marginalizing over A, but the expression P(A, B) does not have a closed form so the integration is tricky.

Any thoughts on how P(X, A, B) can be derived, without discarding information? Many thanks in advance.

Amit

Answer 1

Ok, it has been a long time since I've done joint probabilities so take this with a big grain of salt but the first place I would start looking, given that A and B are orthogonal, is for an expression something like:

P(X, A, B) = P(X,A) + (P(X,B) * (1-P(X,A)));

Again, this is just to give you an idea to explore as it has been a very long time since I did this type of work!

Answer 2

What you're dealing with here is an undirected acyclic graph. A is conditionally independent of B given X, but X depends (I assume directly) on A and B. I'm a little confused about the nature of your problem, i.e. what form your probability distributions are specified in, but you could look at belief propagation.

Answer 3

Your question is very unclear in terms of what you observe and what are unknowns. It seems like the only fact that you state clearly is A and B are independent given X. That is,

Assumption: P(A,B|X)=P(A|X)P(B|X)

Hence: P(A,B,X)=P(A,B|X)P(X)=P(A|X)P(B|X)P(X)=P(A,X)P(X)=P(B,X)P(X)

Take your pick of factorizations.