If the ratio is greater than 1, then the posterior probability of either A or B will be higher than the prior probability of either A or B.
I suppose it's as if A correlates with B.
However, if A correlates with B, then A and B are random variables instead of "events".
So I wonder if we have something like this in probability. Basically, 2 variables, indicate one another.
I wonder if
P(A&B)/P(A)P(B) = P(A|B)/P(B) = P(B|A)/P(A) has a name?
That name, let's call it L, (I think it should be called evidence strength or something) has an interesting property.
It shows how much increase of probability an evidence give and it is always symmetrical.
Imagine a drug test with 99% sensitivity and 99% specificity. Imagine .5% drug users. So out of 100000 people tested we have 500 users. 495 of which is tested positive. We also have 99500 non users. 995 of which is tested positive.
Here, the probability of user is .5% Probability of those tested positive is around 1.5%
The ratio I am talking about is around 66.
If we know a guy is a user, we know how likely he is tested positive. Just multiply the probability of being tested positive by 66 and we get around 33%.
If we know a guy is tested positive, we know the probability that he is a user. Just multiply the probability that he is a user, which is .5% by 66 and we get 33%
That means the strength of evidence of one path of reasoning is exactly the same both way. A->B doesn't always mean B->A. However, they increase the probability of the conclusion by the same amount.