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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.