TITLE: Could someone explain conditional independence?
QUESTION [119 upvotes]: My understanding right now is that an example of conditional independence would be:
If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are independent; that is, we wouldn't expect one to have an affect on the other. But if a snow storm hits the city and introduces a probability C that traffic will be at a stand still, you would expect that the probability of both A getting home in time for dinner and B getting home in time for dinner, would change.
If this is a correct understanding, I guess I still don't understand what exactly conditional independence is, or what it does for us (why does it have a separate name, as opposed to just compounded probabilities), and if this isn't a correct understanding, could someone please provide an example with an explanation?

REPLY [8 votes]: No independence
Take a random sample of school children and for each child obtain data on:

Foot Size ($F$)
Literacy Score ($L$).

The two will be (positively) correlated, in that the bigger the foot size the higher the literacy score.
The random variables $F$ and $L$ are not independent.
Confounder

Obviously a bigger foot size is not the direct cause for a higher literacy score. What correlates the two is the child's age ($A$), which is the confounder in the fork structure above.
If I tell you someone's foot size, it hints at their age, which in turn hints at their literacy score. So we can write:
$$
P(L|F) \neq P(L)
$$
Again, the random variables $F$ and $L$ are not independent.
Conditioning
By conditioning on age (the confounder), we no longer consider the relationship between foot size and literacy for the whole sample, but per each age group separately.
Doing so annihilates the correlation caused by the confounder, and makes foot size and literacy score independent.
While age does hint at literacy score, if now I tell you someone's foot size it doesn't hint a smidgen about their age because their age is given (we condition on it) - no correlation.
$$
P(L|F, A) = P(L|A)
$$
And so:
$$
P(L|F) = P(L)
$$
Conclusion
So this was just an example of two random variables $F$ and $L$ that were:

dependent when not conditioned on $A$
independent when conditioned on $A$

We say that $F$ is conditionally independent of $L$ given $A$:
$$
(F \perp L | A)
$$