WEBVTT
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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.