Does Bayes' theorem have constraints for it to work?

Question

I am wondering if there's any situation where $P(A|B)*P(B) ≠ P(B|A)*P(A)$ ???

I have a scenario, where I want to compute the similarity of two users. And these users can be described with a few features. Say $P(Us)$ represents two users' similarity and they have features $I$ and $T$. I can defined the probability that two users are similar (a.k.a. user similarity) as

$P(Us)=P(Is Ts)$, where $Is$ means two users are similar on $I$ feature and $Ts$ means two users are similar on $T$ feature. So the more two users are similar in each of the two dimensions, the more the two users are similar.

So I calculated P(Is|Ts)*P(Ts) and P(Ts|Is)P(Is) in my scenario but they are not identical. It is a real scenario problem. Yes I know maybe that also depends on how the two ocnditional probabilities are defined in my case. But I am wondering if there's any constraints to make sure they have the identical result?

Answer 1

No there are no constraints. Both sides of the equality

$$P(A|B)\times P(B) = P(B|A)\times P(A)$$

are always equal to $P(A\cap B)$.

Answer 2

By definition for any event $B$ with positive probability $$ P(A|B) = P(A \cap B)/P(B) $$

So if $A$ and $B$ have positive probability it is always true that $$ P(A|B) * P(B) = P(A \cap B) = P(B|A) * P(A) $$