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rLlZpnT02ZU
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So in this chapter, we're
going to talk about maximum
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likelihood estimation.
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Who has already seen maximum
likelihood estimation?
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OK.
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And who knows what a
convex function is?
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OK.
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rLlZpnT02ZU
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So we'll do a little bit of
reminders on those things.
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rLlZpnT02ZU
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So those things are when we do
maximum likelihood estimation,
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likelihood is the function, so
we need to maximize a function.
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That's basically
what we need to do.
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And if I give you
a function, you
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need to know how to
maximize this function.
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rLlZpnT02ZU
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Sometimes, you have
closed-form solutions.
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You can take the derivative and
set it equal to 0 and solve it.
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But sometimes, you actually
need to resort to algorithms
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to do that.
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And there's an entire
industry doing that.
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And we'll briefly touch upon
it, but this is definitely
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not the focus of this class.
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rLlZpnT02ZU
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OK.
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rLlZpnT02ZU
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So before diving directly
into the definition
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of the likelihood and
what is the definition
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of the maximum likelihood
estimator, what
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rLlZpnT02ZU
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I'm going to try to
do is to give you
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an insight for what we're
actually doing when we do
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maximum likelihood estimation.
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rLlZpnT02ZU
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So remember, we have a
model on a sample space E
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and some candidate
distributions P theta.
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And really, your goal is
to estimate a true theta
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star, the one that generated
some data, X1 to Xn,
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in an iid fashion.
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But this theta star is
really a proxy for us
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to know that we
actually understand
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the distribution itself.
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The goal of knowing theta star
is so that you can actually
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know what P theta star.
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Otherwise, it has--
well, sometimes we
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said it has some meaning
itself, but really you
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want to know what
the distribution is.
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And so your goal is to actually
come up with the distribution--
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hopefully that comes
from the family P theta--
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that's close to P theta star.
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rLlZpnT02ZU
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So in a way, what does it mean
to have two distributions that
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are close?
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It means that when you
compute probabilities
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on one distribution,
you should have
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the same probability on the
other distribution pretty much.
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So what we can do
is say, well, now I
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have two candidate
distributions.
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So if theta hat leads to a
candidate distribution P theta
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hat, and this is
the true theta star,
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it leads to the true
distribution P theta star
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according to which
my data was drawn.
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That's my candidate.
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As a statistician, I'm
supposed to come up
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with a good candidate,
and this is the truth.
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And what I want is that
if you actually give me
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the distribution,
then I want when
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I'm computing
probabilities for this guy,
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I know what the probabilities
for the other guys are.
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And so really what I want is
that if I compute a probability
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under theta hat of
some interval a, b,
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it should be pretty
close to the probability
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under theta star of a, b.
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rLlZpnT02ZU
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And more generally,
if I want to take
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the union of two intervals,
I want this to be true.
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If I take just 1/2 lines, I
want this to be true from 0
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to infinity, for example,
things like this.
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I want this to be true
for all of them at once.
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And so what I do is that I
write A for a probability event.
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And I want that P hat of
A is close to P star of A
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for any event A in
the sample space.
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Does that sound like
a reasonable goal
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for a statistician?
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So in particular, if I
want those to be close,
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I want the absolute
value of their difference
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to be close to 0.
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And this turns out to be--
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if I want this to hold
for all possible A's, I
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have all possible events, so I'm
going to actually maximize over
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these events.
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And I'm going to
look at the worst
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possible event on which theta
hat can depart from theta star.
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And so rather than
defining it specifically
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for theta hat and
theta star, I'm
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just going to say, well, if
you give me two probability
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measures, P theta
and P theta prime,
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I want to know how
close they are.
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rLlZpnT02ZU
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Well, if I want to
measure how close they
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are by how they can
differ when I measure
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the probability
of some event, I'm
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just looking at the absolute
value of the difference
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of the probabilities
and I'm just
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maximizing over the worst
possible event that might
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actually make them differ.
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Agreed?
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rLlZpnT02ZU
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That's a pretty strong notion.
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So if the total variation
between theta and theta prime
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is small, it means that for all
possible A's that you give me,
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then P theta of A is
going to be close to P
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