WEBVTT
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Suppose I look at the registry
of residents of my town and
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pick a person at random.
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What is the probability
that this person is
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under 18 years of age?
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The answer is about 25%.
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Suppose now that I tell you that
this person is married.
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Will you give the same answer?
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Of course not.
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The probability of being
less than 18 years
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old is now much smaller.
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What happened here?
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We started with some initial
probabilities that reflect
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what we know or believe
about the world.
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But we then acquired
some additional
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knowledge, some new evidence--
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for example, about this person's
family situation.
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This new knowledge should cause
our beliefs to change,
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and the original probabilities
must be replaced with new
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probabilities that take into
account the new information.
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These revised probabilities are
what we call conditional
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probabilities.
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And this is the subject
of this lecture.
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We will start with a formal
definition of conditional
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probabilities together with
the motivation behind this
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particular definition.
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We will then proceed to develop
three tools that rely
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on conditional probabilities,
including the Bayes rule,
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which provides a systematic
way for incorporating new
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evidence into a probability
model.
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The three tools that we
introduce in this lecture
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involve very simple and
elementary mathematical
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formulas, yet they encapsulate
some very powerful ideas.
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It is not an exaggeration to
say that much of this class
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will revolve around the repeated
application of
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variations of these three
tools to increasingly
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complicated situations.
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In particular, the Bayes rule
is the foundation for the
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field of inference.
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It is a guide on how to
process data and make
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inferences about unobserved
quantities or phenomena.
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As such, it is a tool that is
used all the time, all over
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science and engineering.