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In this lecture, we start
our systematic
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study of Bayesian inference.
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We will first talk a little
bit about the big picture,
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about inference in general,
the huge range of possible
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applications, and the different
types of problems
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that one may encounter.
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For example, we have hypothesis
testing problems in
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which we are trying to choose
between a finite and usually
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small number of alternative
hypotheses or estimation
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problems where we want to
estimate as close as we can an
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unknown numerical quantity.
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We then move into the
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specifics of Bayesian inference.
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The central idea is that we
always use the Bayes rule to
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find the posterior distribution
of an unknown
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random variable based on
observations of a related
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random variable.
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Depending on whether the random
variables are discrete
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or continuous, we must of course
you use the appropriate
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version of the Bayes rule.
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If we want to summarize the
posterior in a single number,
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that is, to come up with a
numerical estimate of the
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unknown random variable, we
then have some options.
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One is to report the
value at which the
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posterior is largest.
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Another is to report the
mean of the conditional
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distribution.
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These go under the acronyms
MAP and LMS.
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We will see shortly what these
acronyms stand for.
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Given any particular method
for coming up with a point
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estimate, there are certain
performance metrics that tell
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us how good the estimate is.
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For hypothesis testing problems,
the appropriate
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metric is the probability of
error, the probability of
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making a mistake.
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For problems of estimating
a numerical quantity, an
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appropriate metric that we will
be using a lot is the
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expected value of the
squared error.
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As we will see, there will be
no new mathematics in this
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lecture, just a few definitions,
a few new terms,
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and an application of
the Bayes rule.
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Nevertheless, it is important to
be able to apply the Bayes
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rule systematically and
with confidence.
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For this reason, we will be
going over several examples.