After reading extensively on the subject I would like to clarify this apparent problem with "Bayes Rule".
Namely the notation often used P(A and B) = P (B and A) has a big assumption that I will try to point out. This might be similar to questions regarding the inverse fallacy that I have seen around but I will try to point out something that I think is more specific although likely related.
Bayes realies on the following statement:
P (A and B) = P (B and A)
and
1) P (A and B) = P(A) * P(B|A)
"this assumes A happens first and A affects B" (i.e. the probability of A is multiplied by the probability of B once we know A has happened).
2) P (B and A) = P(B) * P(A|B)
"again, this calculates the combined probability but in terms of P(B) and the probability of A assuming B has happened (i.e. P(A|B), and therefore assumes B affects A."
This might seem logical as we use this notation because we are dealing with dependent variables. Had we been thinking these variables are independent we would say P (A and B) = P(A) * P(B).
However, my question is "What if the first assumption is correct but the second one is not?" Specifically Bayes seems to assume that if A affect B then B must affect A but this is NOT necessarily true in real life. In this case P(B and A) = P(B) * P(A) and therefore:
P(A and B) ≠ P (B and A)
I can think of several examples in which the order of events have an effect in on direction but not in the other.
So it seems to me that when using Bayes and assuming P(A and B) = P(B and A) we are making a big underlying assumption, in addition to the value estimates that will follow. Some people say well P(A and B) = L (B and A) , meaning that you are using that notation to refer to the fact that you are trying to estimate P(A and B) on the basis of a Likelihood function in terms of B, but then again that approximation could be terribly flawed (just like many things in statistics I suppose).
References: http://www.greenteapress.com/thinkbayes/html/thinkbayes002.html http://link.springer.com/article/10.3758%2FBF03195278
You're bringing causation and temporality in where they don't belong. $A\cap B$ is merely the event which occurs if both $A$ and $B$ occur. There's no implication of temporal order or cause and effect. Events are sets, and $\cap$ has the usual set-theoretic meaning here; hence it's symmetric.