Conditional probabilities are like ordinary probabilities,
except that they apply to a new situation where some
additional information is available.
For this reason, any concept relevant to probability models
has a counterpart that applies to
conditional probability models.
In this spirit, we can define a notion of conditional
independence, which is nothing but the notion of independence
applied to a conditional model.
Let us be more specific.
Suppose that we have a probability model and two
events, A and B. We are then told that event C occurred,
and we construct a conditional model.
Conditional independence is defined as ordinary
independence but with respect to the conditional
probabilities.
To be more precise, remember that independence is defined
in terms of this relation, that the probability of two
events happening is the product of the probabilities
that one of them is happening times the probability that the
other one is happening.
This is the definition of independence in the original
unconditional model.
Now, in the conditional model we just use the same relation,
but with conditional probabilities instead of
ordinary probabilities.
So this is the definition of conditional independence.
We may now ask, is there a relation between independence
and conditional independence?
Does one imply the other?
Let us look at an example.
Suppose that we have two events and these two events
are independent.
We then condition on another event, C. And suppose that the
picture is like the one shown here.
Are A and B conditionally independent?
Well, in the new universe where C has happened, events A
and B have no intersection.
As we discussed earlier this means that events A and B are
extremely dependent.
Within G, if A occurs, this tells us that B did not occur.
The conclusion from this example is that independence
does not imply conditional independence.
So in this particular example, we saw that the
answer here is no.
Given C, A and B are not independent.