WEBVTT
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We now come to the third and
final kind of calculation out
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of the calculations that
we carried out
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in our earlier example.
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The setting is exactly the same
as in our discussion of
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the total probability theorem.
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We have a sample space which is
partitioned into a number
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of disjoint subsets
or events which
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we think of as scenarios.
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We're given the probability
of each scenario.
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And we think of these
probabilities as being some
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kind of initial beliefs.
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They capture how likely we
believe each scenario to be.
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Now, under each scenario, we
also have the probability that
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an event of interest,
event B, will occur.
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Then the probabilistic
experiment is carried out.
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And perhaps we observe that
event B did indeed occur.
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Once that happens, maybe this
should cause us to revise our
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beliefs about the likelihood
of the different scenarios.
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Having observed that B occurred,
perhaps certain
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scenarios are more likely
than others.
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How do we revise our beliefs?
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By calculating conditional
probabilities.
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And how do we calculate
conditional probabilities?
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We start from the definition of
conditional probabilities.
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The probability of one event
given another is the
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probability that both events
occur divided by the
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probability of the conditioning
event.
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How do we continue?
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We simply realize that the
numerator is what we can
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calculate using the
multiplication rule.
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And the denominator is exactly
what we calculate using the
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total probability theorem.
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So we have everything we need
to calculate those revised
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beliefs, or conditional
probabilities.
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And this all there is
in the Bayes rule.
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It is actually a very
simple calculation.
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It's a very simple
calculation.
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However, it is a quite
important one.
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Its history goes way back.
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In the middle of the 18th
century, a Presbyterian
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minister, Thomas Bayes,
worked it out.
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It was published a few years
after his death.
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And it was quickly reorganized
for its significance.
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It's a systematic way for
incorporating new evidence.
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It's a systematic way for
learning from experience.
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And it forms the foundation of a
major branch of mathematics,
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so-called Bayesian inference,
which we will study in some
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detail later in this course.
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The general idea is that we
start with a probabilistic
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model, which involves a number
of possible scenarios.
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And we have some initial beliefs
on the likelihood of
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each possible scenario.
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There's also some particular
event that may occur under
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each scenario.
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And we know how likely it is to
occur under each scenario.
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This is our model of
the situation.
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Under each particular situation,
the model tells us
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how likely event
B is to occur.
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If we actually observe that B
occurred, then we use that
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information to draw conclusions
about the possible
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causes of B, or conclusions
about the more likely or less
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likely scenarios that may have
caused this events to occur.
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That's what inference is.
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Having observed b, we make
inferences as to how likely a
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particular scenario,
Ai, is going to be.
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And that likelihood is captured
by this conditional
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probabilities of Ai, given the
event B. So that's what the
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Bayes rule is doing.
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Starting from conditional
probabilities going in one
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direction, it allows us to
calculate conditional
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probabilities going in the
opposite direction.
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It allows us to revise the
probabilities of the different
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scenarios, taking into account
the new information.
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And that's exactly what
inference is all about, as
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we're going to see later
in this class.