Suppose $A$ and $B$ are two dependent events, that is $P(A\cap B)>0$. We know that $P(A\cap B)=P(A|B)P(B)$. Is it true that $A|B$ and $B$ are independent?
From my understanding, two events $X$ and $Y$ are independent if and only if $P(X\cap Y)=P(X)P(Y)$. It seems to me that $A|B \cap B = A\cap B$, so the joint probability of $A|B$ and $B$ should also be $A\cap B$. Then, they should be independent? Is my intuition correct?
Now, in the context of machine learning, suppose $A$ and $B$ denote distribution of two features for the model. Then, I can empirically calculate $A|B$ from the data. Then if I transform my feature vectors from $(A, B)$ to $(A|B, B)$, they should be independent. Is it actually done in practice?
Edit: Corrected a mistake in phrasing of the question as pointed out in the first answer. (Thanks!) My question is yet to be answered.
No. $P(A|B)$ and $P(B)$ are numbers, not events.