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Maximum Likelihood simply uses all those Max/Min strategies that we learned
in high-school calculus and then promptly forgot.
Here's the general strategy in for solving for the value of a
parameter that maximizes the probability of
the data:

- Take the first derivative of the function with respect to the
parameter that you want to solve for.
- Set the derivative equal to zero and attempt to solve for the
parameter.
- If you come up with a single solution, take the second
derivative of the original equation with respect to the parameter,
substitute in your solution for the parameter
and then check to see that it is less than zero. If so, then you have
found the value that maximizes the function. (This has worked in
almost every situation I have encountered.)
- If you come up with multiple solutions, check all the solutions
and check the endpoints of the range as well. (You almost never
have to do this.)

Often times the log of the likelihood function is maximized instead of
just the likelihood function. This is because it is almost easier to
work with the log of the likelihood function than the likelihood
function itself. We can justify this simplification because all probability distributions are non-negative for the domain of
, and the function is an increasing
function in , thus, the solution for the
parameter that maximizes the probability distribution given the data
is the same as
the maximum of the natural logarithm of the distribution given
the data. Also, we'll use the
notation,
to mean
*The maximum with respect to (the parameter that we
want to estimate) of the probability of the
data, ***X**. It is also worth noting that most
statisticians use ``log'' to mean ``natural log'' or ``ln''.

no_titleno_title

From the overview, let's assume that we have **X**, a vector of
independent
data points,
, collected from the same normal distribution where both
and are unknown. Since each element in **X** is
independent, the probability of the data as a whole is the product of the
probability of each element in **X**.^{3.5}

We will begin by finding an estimate for . To do this we will
assume that we know .

and

The partial derivative with respect to is,
Thus,

Verifying that is indeed a maximum requires us to take the
second derivative of Equation 3.9.1 and make sure it is
negative.

Thus, since is the only extreme point, is indeed a maximum.
Now we will solve for
, the MLE
of . Starting from Equation 3.9.1 and
substituting in our solution for , we can
take the partial derivative with respect to . Thus,

and

To verify that our solution for
is indeed a maximum,
we have,

Substituting in our solution for we have,

and thus, our solution for is also a maximum.
One final note before we conclude this example. If we had attempted
to solve for the MLE for before we solved for ,
then we would to have ended up with the solution

which still contains the unknown parameter . At this point, we
would have to pause in our derivation of
and solve
for . Once we had a solution for , we would
then substitute it in for to complete our derivation of
.

** Next:** Properties of Maximum Likelihood
** Up:** Parameter Estimation Using Maximum
** Previous:** Overview
** Index**
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Frank Starmer
2004-05-19