WEBVTT
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In this segment, we will
go through two examples
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of maximum likelihood
estimation,
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just in order to get a feel
for the procedure involved
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and the calculations that
one has to go through.
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Our first example
will be very simple.
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We have a binomial
random variable
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with parameters n and theta.
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So think of having a coin
that you flip n times,
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and theta is the
probability of heads
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at each one of the tosses.
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So we flip it n
times and we observe
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a certain numerical
value, little k
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for the random variable
K. And on the basis
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of that numerical value, we
would like to estimate theta.
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According to the maximum
likelihood methodology,
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the first step is to write
down the likelihood function.
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This is the probability of
obtaining this particular piece
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of data if the true
parameter is theta.
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Now, since K is a
binomial random variable,
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the probability of obtaining
k heads in n tosses
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is given by this
expression here.
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So what we need to do is to take
the data that we have observed,
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plug it in this formula,
leave theta free--
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we have here a
function of theta--
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and then maximize this function
of theta over all theta.
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Let us now do this calculation.
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Actually, instead of
maximizing this expression,
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it's a little easier to
maximize the logarithm
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of this expression.
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And the logarithm of this
expression is as follows.
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There's a first term, which
is the logarithm of the n
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choose k term.
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Then, the logarithm of theta
to the k is k times log theta.
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And finally, the
logarithm of the last term
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is n minus k, log
of 1 minus theta.
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So we need to maximize this
expression with respect
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to theta.
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In order to do that, we take
the derivative with respect
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to theta.
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Here, there is no
theta involved.
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We get a contribution of 0.
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This term has a derivative
of k divided by theta.
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And this term here
has a derivative,
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which is n minus k
times the derivative
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of this logarithmic
term, which is
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1 over what is
inside the logarithm.
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But by the chain rule,
because of this minus sign
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here, we get also a minus sign,
and we obtain this expression.
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Now, at the maximum,
the derivative
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has to be equal to 0.
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And this gives us now
an equation for theta
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that we can solve.
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Let us take this term, move
it to the right-hand side,
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and then cross-multiply
with the denominators
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to obtain the relation
that k minus k theta-- this
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is obtained by multiplying this
k with this one minus theta
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factor-- has to be equal to
this term times theta, which
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is n times theta minus k theta.
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The k theta terms
cancel, and we're
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left with this
expression, which tells us
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that theta should be
equal to k over n.
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So this is the maximum
likelihood estimate
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for this particular
problem, which
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is a pretty reasonable answer.
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If you would like to
rephrase what we just
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found in terms of estimators
and random variables,
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the maximum likelihood
estimator is as follows.
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We take the random variable that
we observe, our observations,
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and divide it by n.
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And this is now a
random variable,
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which will be our estimator.
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Now, notice that in
this particular example,
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the answer that we got is
exactly the same as the answer
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that we got in the context
of Bayesian inference
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when we were finding the
maximum a posteriori probability
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estimator, but for
the special case
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where the prior was a
uniform distribution.
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So if we assume that theta
is actually a random variable
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but has a uniform distribution,
so that we have a flat prior,
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and we carry out maximum
a posteriori probability
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estimation.
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We do obtain exactly
the same estimate.
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And this is consistent
with the comments
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that we made earlier, that
maximum likelihood estimation
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can be interpreted also as MAP
estimation with a flat prior.
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Let us now move to our
second example, which
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will be a little
more complicated.
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Here, we have n random
variables that are independent,
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identically distributed.
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They all have a
normal distribution
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with a certain
mean and variance.
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But both the mean and
the variance are unknown,
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and we want to estimate
them on the basis
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of these observations.
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The first step is to write
down the likelihood function.
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That is the probability
density function
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for the vector of observations
given some set of parameters.
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Because of independence,
the joint distribution
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of the vector of X's that we
have obtained is the product
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of the PDFs of the
individual X's, of the Xi's.
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So the PDF of the typical Xi
that has variance v and mean mu
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is of this form.
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So this is the likelihood
function in this case.
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This is the probability
density of obtaining
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a particular vector
X of observations
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when we have these
particular parameters.
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We would like to
maximize this function.
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As in our previous example,
it is actually a little easier
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to maximize the logarithm
of this expression.
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And this is the
same as minimizing
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the negative of the
logarithm of this expression.
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Now, when we take the
logarithm of this expression,
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we have a product.
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So we're going to get
a sum of logarithms.
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And I leave it to you to verify
that the negative logarithm
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of this expression is of this
form plus some other constant
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that does not involve
the parameters,
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and which comes from this factor
of 1 over square root 2pi.
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In particular, this
term here appears
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when we take the
logarithm of this.
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And this happens n times because
we have a product of n terms.
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And this term here
appears when we
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take the logarithm
of this expression,
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and after we put
in the minus sign,
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because we're
actually considering
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the negative of the logarithm.
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Now, to carry out the
minimization, what
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we need to do is to take the
derivative of this expression
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with respect to
mu, set it to zero,
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and also take the
derivative with respect to v
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and set it to zero as well.
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Solve those equations and find
the optimal mu and v. So let's
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start by optimizing
with respect to mu.
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So we're going to take the
derivative of this expression
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with respect to mu
and set it to zero.
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This term does not
involve mu, so we only
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need to take the
derivative of this.
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And the derivative
of this is going
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to be-- there's a term
1 over v. And then
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the derivative of a
quadratic divided by 2
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is just xi minus mu.
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And we have one term
for each possible i.
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We get this equation.
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Now we can cancel out v, and
we're left with the equation
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that the sum of the
xi's is equal to the sum
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of the mus, which is n times mu.
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And now we can send
n to the denominator
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to obtain that
the estimate of mu
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is going to be the sum
of the xi's divided by n.
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So the maximum likelihood
estimate of the mean
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takes a very simple
and very natural form.
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It is just the sample mean.
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Now, let us continue with
the minimization with respect
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to v. In order to carry
out that minimization,
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we need to take the derivative
of this expression with respect
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to v and set it to zero.
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The derivative of
the first term is
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equal to n over 2 times 1
over v. And then from here,
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when we take the
derivative, we obtain
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the sum of all these terms
divided by 2v squared.
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But actually, when we take
the derivative of 1 over v,
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the derivative is
minus 1 over v squared.
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And for this reason here,
we will have a minus sign.
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So this is the
derivative with respect
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to v. We set it equal to zero
and carry out some algebra.
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What is the algebra
involved here?
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We can delete this term, 2,
that appears here and there.
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This term v cancels
out this exponent here.
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Then we take this v, move
it to the other side,
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and then take this n and
move it to this side,
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underneath this term.
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And finally, what we
obtain after you carry out
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this algebra is this
expression, that the estimate
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of the variance is some
form of the sample variance
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where we use the
optimal value of mu.
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And the optimal value of
mu we have already found.
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It's given by this
expression here.
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So we obtain a pretty natural
estimate for the variance
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as well by using this maximum
likelihood methodology.
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Now, these two examples
were particularly nice
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because the algebra was
not too complicated.
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And the answers
turned out to be what
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you might have guessed without
using any fancy methods.
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But in other problems,
the calculations
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may be more complicated and the
answers may not be so obvious.