Indeed we have P(A|B) = P(A and B)/P(B), so that P(A|B)/P(A) = P(A and B)/(P(A)P(B)), which is clearly symmetric in A and B, so we get Bayes' theorem.
The reason for the usual form for it is just that we typically want to compute P(A|B) from P(B|A), and the usual form gives this directly.
Whenever I need to talk about Bayes' theorem's derivation (which is somewhat distinct from its application), I always use form - based on the fact that both are equal to probability of conjunction. It is a bit more robust when putting zeroes in, relative to any division form.
I am trying to deducing a system of quatifying the relationship between 2 events(variables) by using Bayes' theorem:
For assessing if event A and event B are related:
P(A|B)/P(A)=P(B|A)/P(B) means P(A|B)/P(A)-P(B|A)/P(B)=0.
It tells the direct relationship between A and B. Thus, in dynamic setting, changes in P(A|B)/P(A) should be the same as P(B|A)/P(B). If they are not the same, then we can say that they are not directly related or may have hidden errors.
if [P(A|B)/P(A)] or [P(B|A)/P(B)]=1, we can say they are unrelated.
And for event A and B, if [P(A|B)/P(A)] or [P(B|A)/P(B)]>1, it means then are positively related, vice versa.
However, if real life, it's hard to get P(A|B) or P(B|A). What we usually got maybe only be their changes, over time or over other parameters.
So I propose, say that the greater the deviation of d[P(A|B)/P(A)]/d[P(B|A)/P(B)] from 1, provided that P(A|B)/P(A)] or [P(B|A)/P(B)]=/=1, the greater the relationship between A and B and vice versa.
And we can also propose that [d[P(A|B)/P(A)]/dt]/[d[P(B|A)/P(B)] /dt] should be proportionate, if in a closed system.
And if d[P(A|B)/P(A)] or d[P(B|A)/P(B)] is 0, it means A and B are unrelated! or at least no more related. In real life, there's so call marginal effect, which can be illustrated by this.
I have a feeling that we can dive further into this kind of things. Any comments is welcomed.
Maybe it can be used to differentiate signal from noise? or detecting any confounders?
What I mean is:
To decomposed Bayes' Theorem into two illuminating parts:
This is Bayes' Theorem in its easier understanding form:
I've got some insight on the Bayes' theorem.
P(A|B) = P(B|A) * P(A) / P(B).
Can't we switch the thing into:
P(A|B) = P(A)*[P(B|A) / P(B)] .
In that way, P(A|B) equals to the P(A) times ratio of P(B|A) / P(B). It means P(A|B) is just the ratio of P(B|A) / P(B) in P(A). It's much more simpler this way.
And from that, we can also change it to:
P(A|B)/P(A) = P(B|A) / P(B) .
Which means the two ratio should be correct. It means this theorem assumes that these 2 ratio should be the same.
I wondered why so many textbooks, YouTube, wiki never explains it this way. It's much simpler this way.