Does $A \perp B\mid C$ implies anything when we already know $A\perp B$?

Question

I am confused with this conditional independence situation. If we already know $A$ and $B$ are independent random variables, is there any point of statement like $A\perp B\mid C$? Does it say anything about relationship between $A,B$ and $C$?

Answer 1

There are situations where $A$ and $B$ are independent, but not conditionally independent given $C$, as well as vice versa.

It is easy to construct a situation where $A \perp B \mid C$: just construct a joint distribution the factors as $p(A, B, C) = p(C) p(A \mid C) p(B \mid C)$. For example, $C$ is uniform on $[0, 1]$, and, conditioned on $C$, the random variables $A$ and $B$ are independent coin flips with probability $C$ of heads. In this example $A$ and $B$ are not independent (why?).

For the other counterexample, suppose $A$ and $B$ are independent Bernoulli random variables, and $C = \begin{cases} 0 & A+B \text{ is even} \\ 1 & \text{otherwise}\end{cases}$. Then conditioned on $C$, knowledge of $A$ will exactly determine what $B$ is, so $A$ and $B$ are not conditionally independent given $C$.