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Suppose I have a fair coin which
I toss multiple times.
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I want to model a situation
where the results of previous
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flips do not affect my beliefs
about the likelihood of heads
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in the next flip.
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And I would like to describe
this situation by saying that
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the coin tosses are
independent.
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You may say, we already
defined the notion of
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independent events.
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Doesn't this notion apply?
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Well not quite.
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We defined independence
of two events.
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But here, we want to talk
about independence of a
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collection of events.
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For example, we would like to
say that the events, heads in
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the first toss, heads in the
second toss, heads in the
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third toss, and so on,
are all independent.
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What is the right definition?
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Let us start with intuition.
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We will say that a family of
events are independent if
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knowledge about some of the
events doesn't change my
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beliefs, my probability model,
for the remaining events.
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For example, if I want to say
that events A1, A2 and so on
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are independent, I would like
relations such as the
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following to be true.
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The probability that event A3
happened and A4 does not
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happen remains the same even
if I condition on some
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information about some
other events.
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Let's say if I tell you that
A1 happens or that both A2
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happened and A5 did
not happen.
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The important thing to notice
here is that the indices
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involved in the event of
interest are distinct from the
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indices associated with the
events on which I'm given some
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information.
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I'm given some information about
the events A1, A2, and
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A5, what happened to them.
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And this information does not
affect my beliefs about
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something that has to do
with events A3 and A4.
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I would like all relations
of this kind to be true.
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This could be one possible
definition, just saying that
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the family of events are
independent if and only if any
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relation of this type is true.
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But such a definition would not
be aesthetically pleasing.
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Instead, we introduce the
following definition, which
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mimics or parallels our
earlier definition of
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independence of two events.
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We will say that a collection
of events are independent if
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you can calculate probabilities
of intersections
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of these events by multiplying
individual probabilities.
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And this should be possible
for all choices of indices
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involved and for any number
or events involved.
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Let us translate this into
something concrete.
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Consider the case of three
events, A1, A2, and A3.
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Our definition requires that
we can calculate the
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probability of the intersection
of two events by
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multiplying individual
probabilities.
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And we would like all of these
three relations to be true,
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because this property should
be true for any
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choice of the indices.
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What do we have here?
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This relation tells us that
A1 and A2 are independent.
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This relation tells us that
A1 and A3 are independent.
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This relation tells us that
A2 and A3 are independent.
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We call this situation pairwise
independence.
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But the definition requires
something more.
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It requires that the probability
of three-way
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intersections can also be
calculated the same way by
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multiplying individual
probabilities.
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And this additional condition
does make a difference, as
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we're going to see in
a later example.
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Is this the right definition?
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Yes.
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One can prove formally that
if the conditions in this
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definition are satisfied,
then any formula of
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this kind is true.
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In particular, we also have
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relations such as the following.
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The probability of event A3 is
the same as the probability of
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event A3, given that
A1 and A2 occurred.
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Or the probability of
A3, given that A1
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occurred but A2 didn't.
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Or we can continue this
similarly, the probability of
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A3 given that A1 did
not occur, and A2
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occurred, and so on.
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So any kind of information that
I might give you about
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events A1 and A2--
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which one of them occurred
and which one didn't--
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is not going to affect my
beliefs about the event A3.
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The conditional probabilities
are going to be the same as
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the unconditional
probabilities.
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I told you that this definition
implies that all
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relations of this kind [are]
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true.
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This can be proved.
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The proof is a bit tedious.
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And we will not go through it.