Assume I have two random variables $A$ and $B$, which are not independent. In my particular case they will be values of a stochastic process at two given points in time, where $A$ is observed at an earlier time.
Define $(B|A)$ to be a conditional random variable, i.e. a random variable defined by the conditional distribution of $B$ given $A$. Question: is $(B|A)$ independent of $A$? Why? Why not? Under what conditions it is?
EXAMPLE: Let $A$ and $C$ be two independent Gaussian (0, 1) random variables, and let $B=A+C$. Then $(B|A=a)$ is Gaussian (a, 1) and seems to be independent of $A$, but I am not sure how to work formally with these kind of things.
EDIT: As Sebastian Andersson pointed out in the comment, $A$ and $(B|A)$ seem not to be independent. However, what if we condition on $A=a$, where $a$ is a constant? The intuition would be that first we are interested in the uncertain event $A$, and then after it happens (and we know the outcome), we are interested in an event $(B|A=a)$. Does it make sense?