While, I was reading up on the Kalman filter, I came across the fact that, "conditional independence doesn't imply absolute independence or the vice-cersa".
i.e:
To see this, let's see a few examples (taken from the internet):
Example 1:
Flip 2 fair coins. Let A be the event that the 1st coin is heads, B the event that the 2nd coin is heads, C the event that both coins are the same(both heads/ both tails). Clearly A and B are independent, but they are not conditionally independent given C. If u know that C has happened, then knowing A tells you a lot about B.
What's interesting here is that A,B,C are pairwise independent but not mutually independent(since any 2 determine the 3rd)
A Counterexample:
We have a bag containing 2 identical looking coins. One of them (coin #1) is biased so that it comes heads 99% of the time and the other (coin #2) comes up tails 99% of the time.
We will draw a coin at random from the bag and flip THE SAME COIN twice. Let A be the event that the 1st flip i s heads. B the event that the 2nd flip is heads. These are clearly not independent.
But let C be the event that coin #1 was drawn (where P(C) = 1/2). Now A and B are conditionally independent given C: if you know that C happened, then you are just doing an experiment where you take a 99% heads coin and flip it twice.
Whether the 1st flip is heads/tails, the coin has no memory so the probability of the 2nd flip being heads is still 99%. So if you already know which coin you have, then knowing how the 1st flip came out is of no further help in predicting the 2nd flip.