I was going through my Probability textbook and I came across this note:
Conditionally dependent random variables may become independent through randomization.
Of course it pertains to the topic of conditional probability but I'm not entirely sure about the randomization portion. Does that just pertain to picking random values?
My guess is that by 'randomization' they mean 'unconditioning'.
The randomization is the 'averaging over' whatever variable you were originally conditioning on.
I guess this is just a caution against assuming that just cause two variables are conditionally dependent that they must be unconditionally independent. Like say $X$ and $Y$ are independent and $Z=(X+Y)/\sqrt{2}.$ Then $X$ and $Y$ are dependent conditional on $Z$. Silly example, but if you turn it around it gets more salient:
Say you are given bivariate standard normals $Z$ and $Y$ with correlation $1/\sqrt{2}.$ Then you say: I'm gonna make a new variable $X$ by taking $\sqrt{2}Z-Y.$ You might be forgiven for assuming $X$ and $Y$ are dependent without a second thought, since you used $Y$ to make $X$, for heaven's sake. However, this is the exact same situation as in the previous paragraph. $X$ and $Y$ are independent when you randomize over $Z.$ In other works if you did the experiment many times and then just looked at the data for $X$ and $Y$, they wouldn't give any information about one another.