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Conditional probabilities are
like ordinary probabilities,

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except that they apply to a
new situation where some

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additional information
is available.

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For this reason, any concept
relevant to probability models

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has a counterpart
that applies to

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conditional probability models.

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In this spirit, we can define
a notion of conditional

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independence, which is nothing
but the notion of independence

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applied to a conditional
model.

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Let us be more specific.

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Suppose that we have a
probability model and two

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events, A and B. We are then
told that event C occurred,

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and we construct a conditional
model.

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Conditional independence
is defined as ordinary

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independence but with respect
to the conditional

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probabilities.

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To be more precise, remember
that independence is defined

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in terms of this relation, that
the probability of two

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events happening is the product
of the probabilities

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that one of them is happening
times the probability that the

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other one is happening.

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This is the definition of
independence in the original

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unconditional model.

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Now, in the conditional model we
just use the same relation,

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but with conditional
probabilities instead of

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ordinary probabilities.

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So this is the definition of
conditional independence.

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We may now ask, is there a
relation between independence

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and conditional independence?

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Does one imply the other?

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Let us look at an example.

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Suppose that we have two events
and these two events

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are independent.

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We then condition on another
event, C. And suppose that the

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picture is like the
one shown here.

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Are A and B conditionally
independent?

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Well, in the new universe where
C has happened, events A

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and B have no intersection.

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As we discussed earlier this
means that events A and B are

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extremely dependent.

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Within G, if A occurs, this
tells us that B did not occur.

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The conclusion from this example
is that independence

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does not imply conditional
independence.

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So in this particular example,
we saw that the

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answer here is no.

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Given C, A and B are
not independent.