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In the previous lecture, we
went through examples of

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inference involving some
of the variations

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of the Bayes rule.

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One case that we did not
consider was the case where

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the unknown random variable and
the observation are both

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continuous.

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In this lecture, we will focus
exclusively on an important

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model of this kind.

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And because we consider only one
specific setting, we will

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be able to start it in
considerable detail.

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In the model that we consider,
we start with some basic

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independent normal
random variables.

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Some of them, the Theta j, are
unknown, to be estimated.

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And some of them, the
Wi, represent noise.

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Our observations, Xi, are linear
functions of these

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basic random variables.

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In particular, since linear
functions of independent

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normal random variables are
normal, the observations are

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themselves normal as well.

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This is probably the most
commonly used type of model in

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all of inference
and statistics.

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This is because it is a
reasonable approximation in

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many situations.

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Also, it has a very clean
analytical structure and a

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very simple solution.

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For example, it turns out that
the posterior distribution of

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each Theta j is itself normal
and that the MAP and LMS

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estimates coincide.

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This is because the peak of a
normal occurs at the mean.

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Furthermore, these estimates
are given by some simple

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linear functions of
the observations.

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We will go over these facts by
moving through a sequence of

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progressively more
complex versions.

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We will start with just one
unknown and one observation

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and then generalize.

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And we will illustrate the
formulation and the solution

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through a rather realistic
example where we estimate the

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trajectory of an object from
a few noisy measurements.